Tone equation. Is it the amp? The guitar? The pickups? The wood?

It is the total package including the player. The player is the most important factor. Exact same gear with a different player will sound different. Next is the amp which includes the speaker. Next is the pedals. Last is the guitar. That said the same player with different gear will sound different because the player is strongly influenced about how they feel about their gear.
 
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These two P90 SGs sound very different from each other. They sounded different when they both had the same bridges too. When both had stock Gibson P90s they sounded different.
Each of the two has its own personality(sound wise). The small guard has the pickups spread wider apart...the bridge pickup is closer to the bridge, and the neck pickup is closer(very slightly) to the neck. This guitar is a bit more brash sounding. It has more contrast in tone, between the two pickups, than the black batwing.
The batwing sounds more civilized, smooth, when changing pickups. The bridge pickup is easy to manage with higher gain.
The batwing, I am comfortable using to play anything in my wheelhouse. The small guard is more like my “punk rock”, edgy, SG...or my wooly, 70s fuzz/muff machine. If I’m attentive with the tone controls, I can keep them sounding fairly close to each other tonally....they’re just two different beasts really...and I’m always surprised as to how different, when I play them back to back. It ain’t the paint that makes the difference.
 
As Alligatorbling says, the amp does seem to be core. If the guitar is set up well with hardware that keeps it in tune and decent puppies then the amp seems to make the most difference (obviously pedals too - stick a fuzz on, and it'll change the sound...).

Many times you hear $100 guitars played well through good amps sounding a lot better than $1000 guitars played well through poor amps (many examples on Youtube).
 
For general consideration, here is my list of contributing factors, in decreasing order of importance:

1. Amp
2. Speaker
3. Pickups
4. Electronics (Pots and Capacitors)
5. Scale length and Construction (shout out to @Robert Herndon . I discovered the effect of scale in my own Jackson mod craziness.)
6. String Gauge
7. Body and Neck Material

But, the most important of all is Mojo, Baby, yeah!

 
The amp will make the biggest difference, plus the basic type of cabinet. I find that any closed back 412 with some speakers that are vaguely close to a Celestion Vintage 30 I can get close enough to my Orange cab for live purposes with a little EQ tweaking (mainly the presence).

Similarly guitars that are roughly in the same ballpark will sound close to each other through the same amp. Hot humbuckers, vintage humbuckers, P-90s, hot single coils, vintage single coils ... as long as the body / neck / bridge are a similar set up you will get CLOSE.

That said, obviously there are small differences that you will notice, and there are benefits to changing pickup models, fitting a brass tremolo block, or whatever. But you mainly notice them in a recording environment. Jam room or gigging not so much. Amps though you really do notice.
 
String tension/weight/tuning and pick thickness probably have more effect than things like the wood. With construction what I notice is a dead sounding guitar versus a really resonant one.
 
I'm going with 80% is in your hands. I have 5 amps I built that I play live. And cut guitars down to 3 I use live.
Robert will say I'm a hoarder I don't have any idea how many guitars and amps I own. I know I built 151 amps and gifted 3
I inherited my brother in laws guitar collection they are still in the shipping boxes and his amp collection is in Washington State
I ruptured a disc in my neck C-6 so I'm not lifting much now and have 5 guitar amp repairs at the music store to pick up today.
Leo said he would load the amps and unload them at my house nice of him.
 
IMO, and purposely bypassing the 'tone is in your fingers' conversation, it's a combination of all factors but to me the biggest and most obvious is the amp and its configuration. Next big one to me is scale length and setup (including fret type and height) of the guitar. Then pickups and signal chain.
 
IMO, and purposely bypassing the 'tone is in your fingers' conversation, it's a combination of all factors but to me the biggest and most obvious is the amp and its configuration. Next big one to me is scale length and setup (including fret type and height) of the guitar. Then pickups and signal chain.

So...if it's the amp in your configuration what is it with a unit that uses powered speakers such as the AxeFX, Kemper or Helix?

I'm running my Helix through a Carvin power amp into a 2x12 cab. So what is my biggest contributing factor?
 
I'm going with:

Hands
Amp
Speaker(s)
Guitar (scale, pickups, wood)

Most everything I play, I sound like me. Even if I use other people's gear. That's why I put hands first. I know if I plugged into David Gilmour's or Michael Schenker's rigs, I would not sound like them. I would sound like me.
 
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