After a ridiculous amount of searching, I finally managed to figure this out. First I found this:

And somehow managed to figure out that the Model 7384 oscilloscope was one of the ones that this camera was made to attach to. Of course the manual didn't say squat about cameras, so I had to read though a bunch of schematic diagrams until I found the one for the power supply that showed the leads connecting to the camera power connector. Finally.... +15V. And the black and white striped wire is some sort of sync line with the oscilliscope that syncs the camera up to take a photo of exactly one refresh ("sweep") of the screen. Not sure I care too much exactly how this works because the only thing I can figure it might be good for is flash sync, and there's already an X-sync connector on the side.

I tend to keep the wall warts from old broken stuff for just this sort of occasion, and I just happened to have a power supply for a broken HP inkjet/scanner that had outputs for +16V and +32V. I tested it with a multimeter and it said 16.5V. Hoping that the camera would actually have a decent voltage regulator (that should be able to deal with an extra 1.5V out of 15V unless it's total crap), especially since this camera used to cost the insane equivalent of $4000 (adjusted for inflation between 1985 and 2012). For that much money I would hope it wouldn't have a total crap voltage regulator.

Unfortunately there wasn't any indication of how much current the camera would try to draw, and this power supply would only output 500mA max, so I kept looking around. Then I found this catalog that actually had more info than the oscilliscope manual did:

This has some interesting stuff in it, but it still didn't say what the current draw was. I guess they didn't have regulations back then requiring that they print this stuff on the chassis somewhere or at least put it in the manual. Well, at least I have an idea of what other cameras they made that might be cheaper or easier to modify.

One thing I did notice is that they used to sell a 15V battery pack that used 12 AA batteries. Given that 12 AA batteries would supply up to 18V if fully charged, that gave me some confidence in the thing's ability to deal with something over 15V. (You have to wonder how long that battery pack actually lasted with this thing.)

Still no idea how much current this thing would try to draw, but I said what the hell and wired up the power supply anyway. I suppose the worst that could happen is that the power supply would blow a fuse, or the easily replaceable fuse in the camera would blow, but it appears to have worked. Either it doesn't draw more than 500mA or it does and the power supply can deal with it anyway. (Unfortunately my multimeter doesn't measure current.)

From what I can gather, the mode settings mean the following:

"NORM" - purely manual settings
"BULB" - shutter is open only while holding the button down
"TIME" - this seems to be a dumb way of saying "bulb mode where you click the shutter button once to open, then click it again to close" (instead of having to hold it down the whole time).
"SINGLE SWEEP" - That's what uses that center pen, or that "gate" BNC connector, to open the shutter for exactly one refresh of the oscilloscope screen. I think the BNC connector is for use on scopes that don't have the 3 pin connector on the front. (The optional battery pack was also sold for the purpose of using the camera on scopes that lacked the power connector on the front.) Anyway, I can't really think of a terribly good use for this.

Not sure what the difference between the "NO" and "NC", but I think they stand for "normally open" and "normally closed". (For example, the NO sync port is only "on" when the flash should fire. The NC is on all the time except when the flash should fire.)

I'm guessing the "remote" port can just be wired to a simple pushbutton switch to operate the shutter as a remote shutter release.

I never did find the actual manual for the C-53, aside from someone selling one on eBay.

Now all I have to do is figure out how I might be able to modify the optics on this thing or otherwise find something useful to do with it.....