I should have been clearer in my description. After noticing that the power
was off to the outlet, I pushed the GFCI reset button but I didn't hear a
click. At that point I figured either the GFCI hadn't tripped and there was
an internal problem, or it had tripped and there was something wrong with
the resetting mechanism. I removed the outlet from the wall and that's when
I noticed the missing ground connection. Regarding the GFCI, there was
nothing wrong with it. I read on a message board that you may have to apply
quite a bit of force to get it to reset. I tried again and it finally reset.
This was the first time this GFCI has tripped.

OK, understood. Yes, those trip/reset arrangements can be "stiff". I
think they may put a lot of tension on the springs inside, to make
sure that the disconnect opens reliably and quickly.

I looked again at the ground wiring you see in the picture. It exits the box
through a hole in the top of the box.

It's fairly common for romex cable (heavy, flexible in-wall wiring) to
have two insulated connectors (hot/black and neutral/white), and one
bare ground wire between them. That's what you may be dealing with
there.

If it's secured to anything outside
the box I can't tell. Is there some official test to perform to check if a
junction box is grounded?

Yes, it can be done, but it's a bit tricky.

The simplest sort of tester is a small plug-like device you connect to
the outlet, which has several LEDs or neon bulbs inside. What you
would want to see, usually, is a couple of lights lit (indicating
that a voltage differential exists between line/neutral and
line/ground), and one or more other lights not lit (e.g. the "hot and
neutral are reversed" light). This will catch a couple of common
wiring faults (no ground at all, and ground/neutral reversal).

These simple testers are cheap, some even come with a "deliberately
trip the GFCI by leaking a bit of current" button, and they're a good
thing to have in your home-repairs drawer.

Gardner Bender GFI-3501 is one such (currently $8 at Home Depot);
there are many other brands.

These won't necessarily tell you that you have a *good* ground on the
circuit, though, because this sort of tester is designed to draw
almost no current at all through the circuit. It may tell you that
ground is connected, when its only connection to the actual building
ground is through corroded conduit and when the "ground" won't
actually carry much current reliably during a fault.

And, it won't catch the case in which ground and neutral have simply
been connected together at the outlet (i.e. no "real" ground). This
is a code violation and can be a safety issue.

Doing a *good* test for these faults usually requires measuring
voltages between the three terminals, while also pulling a substantial
current load on the circuit. Other tests would require powering down
the circuit (e.g. at the main breaker) and measuring resistances.
It's probably best to have a professional electrician perform these
sorts of tests, if you have doubts about the circuit.