There is a lot of debate on this subject but I will try to do my best to answer to this question and explain my position.
Based on my knowledge and experience, there is no audible difference in tone of the guitar because of how electric solid body guitars work.
As you said, in electric guitar, signal is generated in pickups by disturbing its megnetic field with metal strings.
If someone claims that pickups pick any of guitar's body vibrations, try using nylon strigs on electric guitar and you will hear nothing (if you hear anything then your pickups are broken)
Another test is to turn the volume down to zero (or unplug your guitar) and touch something that resonates (and is made of wood preferably), eg. cupboard or a piano with tip of your guitar's head while playing. You will easily hear that while touching the cupboard, vibrations are transfered to it so thus the guitar becomes more audible acoustic-wise. By touching the cupboard, your guitar becomes one with the cupboard and resonates as one big acoustic instrument.
Now, turn the guitar and amp on and do the same thig: play a chord, listen how it sounds and while it is still going, touch the cupboard. You will hear NO difference from the amp, no matter how the guitar sounds acoustically because the sound is coming from the pickups and strings only, not from the wood! Of course, you could try some more destructive tests like chopping off a part of the guitar body (so it affects the accoustic tone, right?) or gluing/screwing the guitar to your desk or floor.
So, how is possible that two guitars sound different? The answer to this question is easy once you forget about all the "facts" that most of the guitar manufacturers say.
First of all: setup - to get similar tone, all pickups need to be set in the same distance from each other and from strings. All strings need to be of the same gauge and type, also in the same distance from the fretboard.
Another, and the most important thing is that you have to know that all passive electronic components (i.e. resistors/pots and caps) are made within a range of tolerance (eg. 1% 5% 10% etc.):
These components are present in your guitar (tone and volume control) and do affect the tone. Also, the wire and magnets used to make pickups can vary to some extent.
This is the factor why the "proof" posted here:
How much does an electric guitar's body physics affect the tone, playability, etc?
is invalid because these two guitars had different electronics (of the same type but not exactly the same).
It would be scientifically better to take two guitars made of different wood, play some chords on one, swap ALL the electronics (including pots), measure the distance between strings, pickups etc, then record some chords, repeat the process about 10 times or so and then do a double-blind experiment.
So, if there is no difference, should all guitars be made of one type of wood (possibly the cheapest one or a plyboard)? I say no. Here's why:
visual difference (a matter of personal preference)
mass (some players prefer lighter, some prefer heavier guitars)
other physical aspects like durability, humidity and thermal expansion of wood
To get a decent sounding guitar, you should worry more how well it is made and not what wood it is made of.