What were the measurements taken when S2 reached periapsis that were the 'best test of GR to date'? There were a few news articles in the popular press and in journals like Science that claimed that the periapsis (or peribothron, as I learned today) of the star S2 orbiting around Sagittarius A* was "the most rigorous test yet" for gravitational redshift.
What specifically were the measurements taken, how much was the starlight redshifted, from which factors? Were the superlatives purely a matter of degree vs kind? Are there some additional effects (in GR or alternative theories) that would only be measurable with the high field? 
A paper by Abuter, et al. about the S2 periapsis mentions

the first-order effects of Special and General Relativity have now become detectable with current capabilities

Were they not before?
 A: I'm not sure I would agree the S2 observations are the best test of GR, or even the most rigorous test. However they are a test in a regime that we have not been able to directly probe before.
The easy tests of GR are all in the weak field limit. For example we can measure gravitational red shift and time dilation on Earth, and we can measure perihelion shifts for several of the planets in the Solar System. These are high precision measurements but measurements on Earth or in the Solar System are measurements in a weak gravitational field near relatively small masses.
We have strong field tests from the LIGO gravitational wave detections, but these are somewhat indirect. That is we assume GR is correct and calculate what the gravitational waves from a black hole merger should look like, then fit those calculations to the LIGO data. The fact we get sensible results is strong evidence for GR, bit it's not as if we directly observed the orbits of the merging black holes.
The S2 measurements give us a direct test of GR near a mass far larger than the mass of the Sun (four million times larger!) and in curvatures that are a lot greater than anywhere in the Solar system, though still fairly small. And these are direct measurements i.e. we measured the orbit directly from the position of S2 and we measured the time dilation by measuring the red shift of the light from S2. And as the articles you linked state, the observations fit perfectly with GR.
So the S2 observations are a very important test of GR because they test it in a regime that had not been tested before. But the best or most precise test? I suppose it depends on what you mean by best or most precise.
