The electrician stopped by and fixed a couple things in the garage, and now it seems to work as expected.

Between when my request for attention and their arrival, I also discovered the outlets didn't work. I plugged the garage door opener into the outlet provided for it, but it wouldn't turn on. I bonked the GFI button a couple times, but it didn't seem to matter. So I plugged it back into the extension cord running from the house.

When the electrician got here, I pointed out the outlet problem, and he also bonked the GFI button. What I thought was just it "clicking into place" was it immediately tripping. He disassembled the outlet and chuckled that it was installed backwards. A couple turns of the screw (and a moment of spark, as we hadn't tripped the circuit breaker...but the spark did), and it was all good again. I plugged the garage door opener into its outlet, and the light turned on, as expected. Power!

He noted that his boss did that one, so he was thankful for a playful razzing he could deliver when they met back at the office.

He then turned to the light switches. He did a few-seconds of flipping switches experiment and found that I was likely correct, as the one switch only worked when the other switch was also on. He took to that with a screwdriver for a few moments, grabbed a little stretch of spare wire, and correctly connected the two switches to be hot all the time. A few switch flips and we confirmed that the exterior light illuminated regardless of the interior lights, and vice versa.

Later, with a bit of chiding from the missus for not checking everything (it was 10°F in the garage, and we were both ready to get to warmer places by the time the wiring was fixed), we noticed that one of the exterior lights wasn't illuminating with the other. We had somewhere to be, though, so I suggested I'd poke at it later (to look for disconnected wires), and drop them a line to come back.

When we got home, and it was dark, we discovered both exterior lights were illuminated. A little poking, and I think it's because the dimmer (it doesn't need a dimmer, but the dimmer remote system was all that was available) wasn't all the way on, and it seems the one light doesn't take to being dimmed very well. It is the case that turning the exterior lights on "fades in," and turning them off "fades out," and the one dims just fine, while the other struggles and doesn't dim well. 

There's no reason to not have the dimmer set to "full on" all the time, so I'm going to dig back into the installation manual to see if I can find the bit I think I saw about setting the dimmer range. Popping on and off is just fine with me.

I am thinking of replacing the interior LED bulbs with some more generous LED panels that screw into the regular sockets. Especially for the loft area, where the single light is over the stairs (as is required by code). We left it to the minimum to install for starts, as I can add another light, outlet, or string some other lights, as we find we might want after we start using the loft.

But, all that to say, it seems like the garage is fully powered now.