What is a Sniper Rifle?

I presume you mean these:

The first one has as more suggestions that the physical layout of the ranges and the bench rests are perhaps better explanations for the situation.

The second page won’t load for me, so I don’t know what it says.

The third is interesting, and includes explanations I hadn’t considered. However, it also says this:

Which means, if I’m interpreting this correctly (and I’m sure you’ll correct me if I’m not), that in absolute terms the bullets don’t get any closer to the target at longer ranges, but in terms of, er… error distance in proportion to distance from gun to target, they are more accurate. So the measurement of the “minute of accuracy” does improve over distance, even though the bullets don’t end up hitting any closer to the target than those fired at shorter ranges.

Am I getting this right?

Yes. That is it exactly. That is what fully stabilized/“going to sleep” means at the shooting range.

I can live with that, though my initial point was made in the belief that “more accurate” meant “closer to the target”, and by that standard I was correct.

In terms of groups on paper, bullets going to sleep roughly translates to the groups at long range being smaller than the groups at close range initially indicate they “should” be. Can you live with that?

Note that there isn’t really a measurement called “minute of accuracy”.

Three different units are in common use by shooters to describe small angles:

  • SMOA = shooter’s minutes of angle = 1" at 100yds
  • TMOA = true minutes of angle = 1/60 of a degree = 1.047" at 100yds
  • MIL (aka USMC MIL) = milliradians = 1/1000 of a radian = 3.438 TMOA = 3.6" at 100yds

Gun makers also sell things they call tactical rifles. Rather than my posting a zillion links most people won’t bother to click, just google image search the term tactical rifle. You’ll see a bunch of different things being marketed as tactical rifles, including some things that you would call a sniper rifle. There are also a few in there that are tactical sniper rifles.
Gun makers give their products names that they hope will increase sales, not necessarily be any kind of useful description.