Fundamentally, it’s more about interpretation. Stress is roughly the ability to avoid serious harm, which applies as much to a hologram as to a flesh-and-blood creature. Injuries represent a loss of function to the point where you’re unable to act, either temporarily (nonlethal) or permanently (permanent, without immediate aid).
From the mid-24th century onwards, tactile interactive holography is enabled by the OHD - the omnidirectional holographic diode - which is a combination of image projector and forcefield generator, which additionally has the ability to remotely produce directional sound. In a contained holodeck environment, this is paired with replicator technology to recreate simple matter, synthesise smells, and produce food and drink. But for an autonomous hologram, it’s just the images, sounds, and forcefield projections to simulate mass.
So, there’ll be situations where an attack or effect that might harm organic crew wouldn’t harm a hologram. Poisons, diseases, and toxins can’t affect a hologram… but some forms of exotic radiation or energy emission might disrupt the holomatrix and cause problems (i.e., damage), and that could include phaser and disruptor blasts through the forcefield.
Further, depending on when your game is set, mobile holoemitters will vary in sophistication. A hologram projected by static emitters might be more able to adjust their matrix to allow objects to pass through them (as the Doctor demonstrates early in Voyager), but the Doctor’s mobile emitter was also a unique invention based on 29th century technology. Attempts to recreate that tech earlier in the timeline might have limitations or simply project less-durable projections. We’ve seen autonomous holograms (as well as for realtime telepresence) in Discovery’s 32nd century future, but never in an environment that lacked emitters, so we’ve got no examples of more-advanced mobile emitters yet.
In combat scenes, treating a hologram more or less the same as an organic character is fine - they’re experiencing the situation differently, but the outcomes are essentially the same, just described a little differently. Attacks can be potentially disruptive to the hologram or the hardware projecting it. A non-lethal injury is the hologram dematerialising temporarily but with no lasting damage to the hardware or software. A lethal injury represents serious damage to hardware or software (or both) which requires immediate attention to avoid losing the program permanently. And yes, using engineering and holography-related focuses to ‘heal’ holograms makes sense.