The Resonance Theory of Magic

In my opinion, this model and the resulting theory reconcile RAW and RAI (and answer a whole bunch more questions to boot). Whether or not it is a true model is irrelevant--it's a useful model that allows predictions.

Postulates

The resonance theory of magic and magical effects attempts to explain the differences between spells, spell-like effects, and those abilities that, while supernatural, are not affected by antimagic fields or similar effects. It relies on 5 postulates (assumptions and definitions). They are:

Postulate 1: 

Surrounding and permeating all of reality is an ambient field of energy. This goes by different names (the Weave is the default in 5e)--for the purpose of this discussion I will call it the aether. 

Discussion: While different areas have different concentrations, very few areas (dead magic zones) are completely lacking this energy.

Postulate 2:

Some individual creatures can concentrate this aether within themselves and use it in various ways. I will call this innate magic--any action that is impossible for a "normal" person in the setting is an example of innate magic. 

Discussion: This includes such "non-magical" effects as a barbarian's Rage, a fighter's Action Surge, or a rogue's Evasion. Using these abilities sometimes temporarily exhausts the store of concentrated aether, requiring a period of rest (albeit small) before they can be re-used or limiting the number of successive uses. In other cases, the changes wrought by this innate magic are permanent and inexhaustible. 

Postulate 3:

Some individuals (not necessarily the same as those in postulate 2) can manipulate aether by imposing a resonant pattern on it, using stored, quantized personal energy Call these resonant patterns spells and the personal energy spell slots.

Discussion: For minor effects or for very practiced casters, the energy needed to induce these resonances is small enough that it's not worth tracking in game mechanics. These are at-will spells (e.g. cantrips). Spell slots are quantized in non-integer ratios--a second-level slot is not exactly twice as big as a first-level slot and they cannot be spent in pieces--either an entire slot is spent or nothing at all. Up-casting a spell (casting it out of a higher level slot than the minimum required) pumps more energy into the resonance which, for some spells, heightens the effect. Certain spells require a measured release of energy into the resonant pattern--maintaining these requires concentration and the biological necessities that imposes, namely that only one spell or spell-like effect can be concentrated on at a time. Restoring spell slots generally is a slow process requiring sleep, as concentrating aether to refill these slots takes time and the soul's full attention. Practiced casters can sometimes hasten that charging process consciously. While different classes differ on how they learn the resonant patterns (and which ones they can learn), all spell slots are identical.

Postulate 4:

Enchanted items resonate with the aether in a similar manner to spells.

Discussion: The source of the energy used to induce this resonance varies--some from stored, concentrated aether (potions, items with charges), some from the wielder (items that are constant effect). Note that charged items seem to regain their charge only when attuned to an individual--otherwise they'd always be found at full charge. This is because they draw on the wielder's soul to concentrate aether. Some items (non-attunement items) only need to be wielded to operate (as their aether draw is de minimis).

Postulate 5:

Some abilities of innate magic manipulates the aether in a spell-like resonant fashion. This is often specified by using the name of the emulated spell in the description.

Discussion: Other innate magic does not operate against external aether but instead is used inwardly to enhance the body or exhaled in a burst of concentrated elemental force, like a dragon's breath.

Results

Given these postulates and definitions, spells and magical effects are those resonant manipulations of ambient magical energy. They're patterned, and may or may not be of extended duration or cover an extended area. The important thing is the resonant pattern. If that pattern is disrupted, the spell effect ceases (at least for that area).

Result 0: Spells do only what they say.

Fireball doesn't conjure a piece of elemental flame, doesn't heat the air to burning. It creates a resonant effect within the area of effect that creates the same end result. Ray of frost can't freeze a pool of water, because the cold isn't there. It's a resonant effect that fades once the energy disperses. Etc. These may or may not follow the usual laws of physics, but certainly one cannot import physical reasoning unless the spell specifies so. One also should not reference another spell to understand a given spell unless the first mentions the second specifically or by mentioning a larger group of spells (a school of magic, for example). Each spell's text stands alone and should be examined with only the general rules as certain reference.

Result 1a: Counterspell

Counterspell is a counter-resonant pattern. It's a momentary, localized noise burst that takes effect quickly, preventing the resonance from forming if successful. Higher power spells are harder to jam, thus requiring either skill (spell-casting ability check) or more raw power (a higher level spell slot) to counter.

Result 1b: Counterspell and non-spell effects

Counterspell only works on actively-produced resonant effects (i.e. spells) as they're taking shape. You can't counterspell a dragon's breath weapon. That's not a resonant effect. You also can't counterspell an existing effect. That's what Dispel Magic is for. 

Result 2a: Antimagic Field

Antimagic Field creates a persistent cancelation field. Any effect that requires a sustained resonance (like fireball, or a magic item) to function is suppressed since the resonance cannot take place within that area. One way of thinking about it is an adaptive damping field (like active noise cancellation) that imposes a pattern that is the inverse of the attempted resonant pattern (effect). This is both complex and requires high power, so it's an 8th level spell that has a limited reach.

Result 2b: Antimagic Fields and innate magic

Innate magic (like non-spell ki, or a dragon's flight, or a dragon's breath) does not require a resonance pattern. Thus it is unaffected by an antimagic field (since damping resonance does nothing against a purely physical, natural effect such as ki-fueled punches. The punches are not resonant, nor is there resonance involved in throwing the extra punch (flurry of blows). The monk is drawing against personal energy to temporarily exceed the normal limits of his body. Dragons keep flying (and breathing) and their breath still works (since the conversion is wild and non-resonant). Barbarians can rage just fine--while this draws on innate magic (because it's not possible for a "normal" person), it is not a patterned effect.

Result 3: Dispel Magic

This is counterspell, except for existing spells and their effects. This pattern disrupts the resonance of stable spells, causing them to end early. It has no effect on innate magic, since it only affects resonance effects.

Oddity Result 4: Detect Magic

This spell does not specify what qualifies as "magic" to be detected. Certainly active spell effects and enchanted items should count (by common usage, if nothing else). If it detects all magic, then everything should light up--the aether is all around and through everything after all. That's absurd on the face, so it must be less than that. So what is it? In this DM's opinion, it should only detect organized, resonant magical effects--items and spells. Since I'm generous, I'll even allow the detection of residues--the after-effects of a major spell or ritual. But that's not clear from the text and so remains an open question. 

Edit: The sidebar (The Weave of Magic, PHB page 205) clarifies that detect magic looks at the ambient field looking for active effects. So this confirms the theory--spells and "magical effects" are all active resonances in the ambient magical field (the weave, the aether, whatever you might call it). 

Magic levels--not one-dimensional 

A constant source of disagreement and dissonance is the level of magic in a setting or system. Settings are often described as "low magic" and then have magic items everywhere; systems like D&D 5e are billed as handling "low magic" settings but have spell casters everywhere. So what does it mean to be "low magic" or "high magic"? I suggest that low or high magic isn't a one-dimensional spectrum. Instead, there are many different axes on which to measure the presence of magic.

The energy that fuels magic and other "supernatural" events is the same energy that makes up the souls of living beings. Aether. Aether is formed within a soul when the soul creates new memories, grows, absorbs information--in general, through the process of creation. When a living thing dies, its aether passes into the Shadows (for intelligent beings) or back into the Prime plane (for plants and lower animals). That energy is pumped up into the astral by the Great Mechanism and passed among the gods and used to maintain the universe itself.

Living spell-casters accumulate aether in knots in their souls. These knots come in varying strengths--two lower level "slots" do not equal one higher slot. These knots are used to fuel patterned resonances with the ambient aether--these patterned resonances are spells. Some spells (cantrips) can be cast solely with the caster's mental energy and do not use up these knots. Without special training, these slots reform over time but require a night's rest to be useable again. There are rumors of some who can force energy into their knots faster, but those are not widespread.

As mentioned in the article Where does magic come from?, the source of all magic is the energy of the soul itself. This article will describe the sources of the patterns ("spells") of each of the traditions of spellcasting ("classes" to use game terms). Although for the sake of the game spells are codified and the same spell cast by casters of different classes have the same effects, in-universe the patterns differ and the effects can be varied. Not all spells in the printed materials can be learned; there are innumerable others that can be cast but are not described in the printed books.