But I still don't completely understand what is or isn't ullage in rocket science context.
Ullage technically is the space in a tank of liquid which is gas-filled instead of liquid-filled.
For a propellant tank, it's important that the ullage volume be kept away from the inlet that leads to the engines, because you want the liquid going into the engines. Propellant turbopumps will surge and likely fail if they ingest large amounts of gas while running; if the gas is inert, pressure-fed engines will lose their stable combustion.
Is it only a noun, or does it have a form as a verb?
Formally it's a noun; as with all jargon, people will in practice reduce complex issues and concepts to a single word without really being conscious of it. Like so: "The Apollo CSM sometimes needs to perform a burn with the RCS thrusters before igniting the main engine, so that it can be ensured that the ullage is at the forward end of the propellant tanks and the propellant feeds correctly" becomes "Apollo does an ullage burn [i.e. a burn which is significant to the ullage issue] before ignition".
In the Apollo transcripts, when mission control reads up a "pad", or set of instructions for a maneuver to be done at a set time, they specify the ullage details like this one from Apollo 11:
73:59:46 McCandless: Roger. TEI-1, SPS/G&N; 38658, minus 0.54, plus 0.65, TIG, 078:02:03.45; plus 2918.0, plus 0377.9, minus 0132.5; roll NA, pitch 041. The balance of the PAD is NA. Ullage; two jets, 19 seconds. TEI-4 PAD, SPS/G&N:, 38658, minus 0.54, plus 0.65, TIG, 084:29:50.59; plus 3137.3, plus 0376.0, minus 0096.8; roll NA, pitch 034. The rest of the PAD is NA. Ullage; two jets, 19 seconds. Both of these PADs are for an undocked maneuver. TEI-plus-4 PAD assumes no LOI-2. Over. Make that TEI-4 PAD assumes no LOI-2. [Long pause.]
But again, ultimately, "ullage" is used here as shorthand: "the pre-burn RCS burn that takes care of the ullage problem".
does ullage always refer to a force that pushes liquid fuel to the end
of a tank where the outlet exists?
Organic Marble notes that the word for this is "settling". "Settling force" is the force that pushes propellant to the inlet end of the tank. "Settling burn" is the maneuver that applies settling force. "Ullage" sometimes gets sloppily used in that role, but the force is either gravity or rocket thrust, not ullage.
Can gas pressure in a tank also containing liquid propellant provide ullage, or is ullage restricted to an acceleration (e.g. from the use of an ullage motor just prior to engine ignition) or a diaphragm or gas bag?
Gas pressure by itself doesn't address the keeping-gas-out-of-the-inlet issue, but a volume of pressurized gas in a tank in microgravity is, technically, ullage. The ullage is moving around in the tank in an uncontrolled way, which is problematic.
The diaphragm/balloon solution to the ullage problem (which is typically found with tankage for small thrusters like the Apollo RCS thrusters) is a terminological edge case. Is it ullage or is it not? It's ullage if you consider both sides of the diaphragm, I suppose, but if you consider the two sides of the diaphragm as separate tanks, then one is a gas tank with no liquid, and the other is a liquid tank with no gas, so there's no ullage in sight.
Regarding sloppy jargon, you may note that in the beer sales business, ullage winds up with the exact opposite meaning -- it's the residual beer in a container after you've emptied it as much as you can, that's going to go to waste rather than being sold. That is, "ullage" means the beer underneath all the ullage!