So, I traced the other lines in the T-fitting in the picture. Indeed, the passenger side line of the t-fitting goes up by the air filter, and the driver's side line is the air/hydraulic cylinder breather line.
That's the vent for sure. I can recall a senior member of this site (rest his soul) referring too that chunk of plumbing as a "common vent", and while not correct, it was very clear to all members what he was referring to.
I remain unsure of what to do with the line that's hanging off the top of the tee.
Attach it correctly (I'll get there), or plug it with a standard (non-"T" union, a plug at the union, or a cap on the line. All can come from the hardware store, there is no pressure to require any consideration here.
I've been looking through TMs, and either I'm missing something in the 250 pages written on brakes, or they just don't have what I'm looking for.
You're missing it. You can (and should) read, read, and read again. The way the information is presented is at such a "step by step" level, that the big picture of the overall assemblies and systems is easy to loose. Your question is more of a "big picture" issue than a pinpoint repair. (Although the "fix" is most definitely a pinpoint fix). Trouble is, the manual can't tell you what to do when somebody's modified the system, so you have to know ALL of that system in order to spot something out of place like you have.
All I see written is whenever work on the master cylinder is done, you start by unscrewing the "hydraulic line" from the elbow on the cap.
It's called that more than once. "hydraulic line" is a poor choice, although in another perspective it is proper. The tubing is multipurpose and often used for fluids, and the fittings primarily used on hydraulic applications, so the assembly (tube and fittings) is called "hydraulic", even though it's use here is not hydraulic.
I must have the old style cap, so I'm assuming I would ignore this.
No. The little plug on your cap is a unit level retrofit. It's been done very often, but it comes at a cost. Searching this site will find discussion of that, but here's the short version. The cap (if working) will allow the pressure inside the brake hydraulic system to drop below atmosphere. The cup seals only seal in one direction. What ends up happening is with sudden temperature shifts it can actually draw miniscule amounts of air into any or all wheel cylinders and the master cylinder fluid area, which builds up over time to give a mushy pedal. You really should have a vent plumbed in correctly. Unscrew that little mushroom cap assembly from your master cylinder cap, and screw in a pipe thread to brake line adapter. Check the threads (or bring it with you), the post above suggested 3/8 pipe thread, and while I surely can't argue I am pretty confident that 1/8 pipe thread and 3/16 tubing is how mine is equipped...
Sooo, if I got the threaded elbow for the MC cap, would the line that's hanging there go into it and voila? Or... does that line need to be removed and plugged and the MC venting done some other way?
Exactly. That line goes directly to the elbow in the master cylinder cap. It should be verified as being able to breathe in case bugs made their nest there, but yeah, it's just as easy as a pipe to brake line adapter, and you're done.
Or, go with the remote reservoir. I've given my thoughts on those in a couple of the more recent posts on the subject. Very worthwhile in my estimation if you're confident in your ability to do so. Just convert that cap to a nipple or tube fitting, run your hose or tube to a reservoir, and that's that. Make sure it's higher, and gravity will take care of the rest. Air and or bleeding does not apply to that part of the system, your master cylinder might bleed it's self by gravity, but if not, it's supposed to have an air space in it anyhow.