I've spent some time with each variation they all have their own pros and cons. I think wrestler is the best IMO let me explain.
Wrestler
Pros
Gotcha Grab is safe -4
Quad grab - great mix up tool (more on this later)
Energy wave pressure (last time i checked +8 on block)
Highest damage in the corner of all variations ( 1 bar 49-51% off of overhead or low) (ending combos in quad grab gives more damage then gotcha grab)
12D3 String ( feint can be canceled into quad grab)
can combo off of ground pound from near full screen. (this is hard)
Can combo after EX Gotcha in corner
Cons
Air grab... kind of pointless
F3/4 into grabs strings are also kind of useless F3/4 have better uses in this variation (more on this later)
Pumped Up
Pros
Improved gotcha grab does more damage then regular gotcha grab/ leaves opponent standing for potential mix up
Full screen Ground pound ( great for zoning/ counter zoning)
Air ground pound (great for zoning/counter zoning, and can do some tricky setups with this) (more on this later)
Energy wave pressure
Cons
Gotcha grab unsafe -8 or 9 if I remember correctly (I think this is huge)
cannot combo off of ground pound
Cannot combo after grab
Heavy Weapons
Pros
Law rocket (great projectile can be aimed, great chip on block, great push back on hit, cannot be ducked)
Law rocket run cancel ( great for keeping up pressure)
Gotcha grab is safe -4
Machine gun ( risky but can beat out armored wake ups if timed correctly)
Can combo off ground pound near full screen (this is hard, seriously)
can combo after ex grab in corner
Cons
Loses energy wave pressure (meh not too bad since you can do law cancel instead)
I made this list off the top off my head so sorry if I forgot something but for the most part this is what each variation has going for it. Let me start off by saying that all 3 of these variations are great and it depends on your play style, I may main wrestler but I have gotten destroyed by pumped up and heavy weapons and vice versa.
It all depends on how you like to play, IMO wrestler is the most offensive of the 3 just because I feel people are not giving the quad grab the respect it deserves I think the move is great how it is, it DOES NOT need flat damage, the timing for each slam is FINE I rarely ever miss it in a decent connection, just practice it and you will be able to do it with your eyes closed. (not kidding I do it alot). Oh and another thing please DO NOT write off the quad grab just because you don't get the last slam. The last slam is purely there for hype purposes, it adds 1% and although it does NOT cost an extra meter its not the end of the world if you just end it on the second to last slam. Like really guys its 1% who cares. This variation gives you a whole extra layer to your mix up they NEED to worry about the grab. And as for a comment I saw above about when to land the grab there are so many different ways to do it. IMO this is one of the easiest grabs to land right next to sun god kotal khans grab.
The list is as follows:
Canceling a BLOCKED F3/4 into quad grab is damn near guaranteed
After any blocked string that leaves you at 0 (11,F21, ect) ( recommend using ex if you wanna do it like this)
Canceling the 12D3 feint into grab ( I use this the most)
Tick throw (after a blocked D1/3)
If you have balls walk up and just do it.
You can also end you combos with quad grab ( I recommend doing this whenever possible)
Again I think this grab is amazing what other grab in the game has so many ways to land, AND lets you decide what side you want the opponent to end up on, AND leaves you close enough for another mix up.
As for pumped up I think its the happy medium, you can keep up with zoners and still have a solid offense off the enhanced grab.
The sneaky trick you can do in pumped up is doing an instant air ground pound for some nice guaranteed damage, they see you jump and usually don't react to it. (saw CDJR do it first so I stole it)
Heavy Weapons gives you the best zoning options and some decent offense with run cancel pressure.
At the end of the day its up to you and your play style I like goin HAM so I use wrestler. Then there is the question of match ups and those do differ but I'll save that for another thread (I will say Pumped up is best variation against grandmaster sub)
I wrote this long ass post just to say, as far as Jax goes there is not one "best" variation. Use the one you like the most they are all really good.
I guess what I am trying to say is
PLAY WHATEVER THE FUCK YOU WANNA PLAY.