What happens when using a GFCI Tester on a non-GFCI outlet

As well it should.

Normally, loads are connected between hot and neutral. Appliances are not supposed to connect hot or neutral to ground; ground is only a shield.

The GFCI compares current on the hot and neutral wires. They should be the same. If they are not, current has found another route, possibly through the grounding system (which isn't supposed to happen) and potentially through some poor human.

Circuit testers are trying to test whether ground is connected... cheaply. They mis-use "hot" as a power source, by connecting a light bulb between hot and ground. If ground is connected correctly, this will light.

In other words, it intentionally creates a hot-ground fault (by sticking a light bulb there). This is exactly the condition GFCIs are designed to detect.

I'm not talking about any ground-fault-test the tester may also have.

So why do testers often work? With a perfect GFCI, they wouldn't work. I suspect it is because GFCI's have detection thresholds above zero, and that is often enough for these testers to "get away with it". I even think there may be a tacit agreement among manufacturers for this, but obviously, lower sensitivity impinges safety. Remember a shock which only stuns you can kill you with secondary effects like falling or drowning.

So either your GFCI is pretty good, or your tester is pretty bad.

My suspicion: an open ground

This is based on two factors:

1. Your three-lamp tester is unable to trip the GFCI, but the GFCI tests OK. This is because the internal TEST button doesn't rely on the ground (it goes load-hot to line-neutral or vice versa), but the three-lamp tester's TEST button is connected hot-to-ground, and thus requires a solid ground connection to work.

2. Pushing the TEST button on your tester yields a "hot/neutral reverse" indication despite it testing correctly initially -- the "correct" indication is due to capacitive coupling between neutral and ground providing a current path "home" for the current through the right-hand lamp.

I'd check to make sure the green wire is securely connected to the box in your case; if it is, then the problem's upstream somewhere as it appears this receptacle was wired using conduit (note the locknut and fitting at the bottom of the box). If you can't find it still -- call in an electrician, as they should have better tools (such as an earth fault analyzer/loop impedance tester) that can be applied to the problem.