It may be different in the fighter world. But in commercial we are trained pretty much to take almost anything into the air once in the high speed portion of the take off.
Outside of serious motor problem what else do the fighters consider a high speed abort for?
Do you guys practice pitch and power? If all instrument quit? I had to do that once in the sim. They failed everything except the engine gauges and standby adi and made us fly to vmc conditions.
For birds like the F-16, F-35, etc.....an engine problem on takeoff is pretty self explanatory in that you aren't going anywhere airborne, or going anywhere far for that matter.
On a jet like the A-10, I was pushing a ~45,000 lb TOW jet with two 8,900 lb/thrust motors, so loss of one on takeoff......a PA-44 had a better single engine climb rate, as I wouldn't have time for fuel dumping, and stores jettison wouldn't buy me too much more VVI.
To your question, it depends on the aircraft. There first needs to be a clear brief of what will be a high speed abort and what won't be. For the F-117, there was
one particular emergency on takeoff.....dual generator failure with takeoff into IMC....that would
need to be an abort, even if high speed. Because that one emergency, simply due to the number of items lost and the number of items you are left with, was a
guarantee that you will lift off to your death or to an ejection scenario; you would
not be coming back with the jet, for a variety of reasons. And even if you did manage to get airborne with the minimal systems available, you'd better be back on the ground in about 10-15 mins, or otherwise you will lose the jet again and will be coming down by parachute, after the battery fails. But that was systems knowledge, and quick assessment skills requirement.
It just depends by aircraft, conditions on takeoff, what the EP is, etc. Its not good to get airborne in an aircraft that you know won't fly or won't fly for long (onboard/cabin fire, structural failure/damage, etc) and end up taking off to your own death; and neither is it good to do a high speed abort for a warning light or master caution, crashing off the end of the runway and killing yourself, and it turns out to have been an antiskid light or some other nuisance light.
The worst one for me personally, would be a scenario where I got airborne in a sick jet that wouldn't fly or fly for long, near a populated area, that I either ended up riding into that area, or jumping out of it and it heading into that area. To me, as a pilot, I not only have a responsibility to myself and any pax I would have onboard, but I also have a responsibility to those on the ground who will be injured/killed were my plane to be dropped into their neighborhood or place of work, etc. Those are things I would consider also, depending where I am, on what I may accept a high speed RTO for in one place, vs where I may make a go decision in others. To me, that comes under airmanship.
In my F-117 example, that emergency I speak of, it was an abort even if past V1, merely because of the
guarantee that you'd be taking off to your death or to someone elses death, or to a parachute recovery at best. The difference there, was that we had the luxury of a drag chute and an arrestor hook, and at some places, a full barrier raise at the departure end. That's not an option for many, I understand; but my primary point is knowing your aircraft, it's systems, what constitutes a no-kidding "I'd better not lift off with this emergency", and a good pre-takeoff brief based on conditions being encountered for that takeoff.
There are indeed some emergencies that are best kept as ground emergencies, and others that are better handled as air emergencies.......depending on a whole host of factors in the mix.