sounds like the hydro-electric pump (that powers the convertible top)should be there by your description. Look around in all of your parts boxes for something that looks like a starter motor (or generator) housing with a domed cap mounted on one end, held on by a bail-clip. It will also have anti-vibration rubber stud-mounts. I'm reasonably sure this is what goes there, again, I'm a V-8 idiot savant, but have spent time around the 51-54 models.

Jeffery--I work in an automobile museum, and the one axiom we have about labels here is that "those who should read the labels won't(the general public), those that do are experts in one make,model and year, and they want to find all of your mistakes". No museum official wants to misrepresent objects on display. The best method is to aviod "catch-all" books like the "standard catalogue of American cars 19XX-19XX" (good books, but no book containing that much data can be 100% correct) and instead get all of your facts and figures from period sales brochures from that manufacturer for that make, model and year. Sales brochures are printed later than salesman's data books and tend to reflect last minute production changes. It also helps to have more than one person double check the text.

OK--sorry I'm a little late on this, you may have already found your problem and fixed it, but here goes anyway.
Lets start out by sorting out some terminology: when you say "short" do you mean "dead short to ground"--where 12V current is literally touching ground causing the 30amp (push button system)circuit breaker on the fender well to click on and off repeatedly, or does the system just not function, stuck in one gear?
When you say "hot" do you mean thermally hot, as in "hot to the touch" (the wire is actually getting hot) or are you using the term "hot" to indicate the presence of 12V current? (A common shop slang term, I use it a lot myself).
When you say that you have conductivity between all of the wires indicated on your post do you mean that with an ohmmeter you get 0 ohms between brown (brown is connected to the reverse gear finger) and the other indicated colors?
I'm going to assume that the unit is not shifting, no one speaks of their pushbutton system until it stops working!
The unit seems from you description to be stuck in neutral (not the worst gear to be stuck in, believe me). You have depressed the "reverse" key on the control panel (shift buttons on the steering column) this has sent 12V current to the reverse finger on the segment switch inside the actuator. The shorting bar of the contact segment is positioned such that the 12V current being applied to the reverse finger is also feeding back power to the white(high),blue(drive), and purple(low)fingers/wires. The switch contacts for all wires/gear functions are "open" (not connected to anything at the moment) at the control panel, by design, so that explains the conductivity you are seeing. One wire you did not mention is the pink wire, and this is where the trouble may be. The two outermost fingers on the segment switch are the leads that go to the motor control relay on the inner fender wall. The pink wire pulls in the relay for one direction of rotation, the orange/black wire the opposite direction. If you have power at the above listed wires and NOT pink, you could have a bent finger for the pink wire (the finger at the left side of the contact segment) OR: it could be fine and the coil for that side of the relay could be bad. Can you push "park" and get any action or is it just dead?
The worst case scenario on these is that one of the two outer fingers (pink on the left, orange/bk on the right) is bent. When this happens you ain't going nowhere. Power can't get from the key on the console to the relay coil to cause rotation.
If you need to see the fingers down on the actuator, disregard the "actuator removal" instructions in the shop manual. Do this:
you need:
A floor jack
Two good hefty jack stands
Normal mechanics tools, sockets, wrenches
A small scissor jack out of a little cheap car
Many varieties of right-angle screwdrivers


*raise the front end (or side if you prefer), put the car on the jack stands

* put the floor jack under the pan of the trans, protect with a scrap of plywood

*remove the rear trans support crossmember, supporting the trans as above

*insert the 'lil scissor jack between the frame and the tail stock of the trans on the pushbutton actuator side (driver's side) and push the trans (diagonally) away from the frame

You can't push the trans too far, there are "stops" sort of built in. This will give you enough room to get the cover off of the segment switch portion of the actuator and see what's up (this is where the trouble is most of the time)

DO NOT REMOVE THE MOTOR FROM THE ACTUATOR UNLESS YOU ABSOLUTELY HAVE TO. DON'T DO IT, DON'T TRY--98% OF THE TIME THE MOTOR IS FINE, IT'S RARELY WHERE THE TROUBLE IS.

When I service a P-B unit this way I replace the slotted screws on the segment switch bridge and the cover with Allen head cap screws. G__ forbid I have to go back in (I usually don't) The allens give you six positions to grab as opposed to two with the fillister head screw.
You may have a situation where the end of the reverse finger is touching the ground of the inner cover (inside the actuator) I have seen this, I have seen contact fingers glow red like a stove burner! this will distemper the contact finger and make it worthless. Follow the above procedure and get a look at what's going on inside the actuator.

Hey, how can I officially join the club?