From what I understood, the 5153 does only one of these at a time, not both.
Doh, you're absolutely right. I looked again at the pictures in the OP, this time pulling them into an image editor and tracing out the rectangles created by the "Even all around" border and the "it looks too tall" border and the one that creates a 1.33 rectangle is the latter (which is the shape necessary for the 5:6 PAR to be correct.). Aspect ratio of the first one is even worse than 1.37, it's more like 1.43. I guess I was just sucked in by how nice it looks that way into thinking that's "correct". (And also I was under the impression that the rounder circle in the test pattern was being displayed with the tighter border, and, well, it looked pretty round to me. Maybe I just can't tell the difference between exactly right and 3% off, while the 15%-ish off 1:1 pixels is obvious.)
(It is notable that you can actually see a *little* bit of the inactive area in the upper left-hand corner of the tighter picture so maybe it should have the dial spun just a tiny bit more, making the "even all around" height more like 1.35-1.40-ish?)
Since BASIC was developed by MS/IBM during design of the 5150 PC, and BASIC was installed in the ROM of every PC, I think there would have been close contact between the software and hardware developers.
So in my opinion, BASIC is probably 'the absolute truth', regardless of how many programs do it differently.
Mind you, BASIC was written a few years before the 5153 was on the market.
Therefore... the 5153 is doing it wrong, or at least its bezel is the wrong shape for looking quite right when it actually doing it right, for the chosen-in-its-ROM PAR corrective ratio of 5:6. This could mean one of two things:
#1: IBM put a great deal of thought into it, decided that there was some compelling reason to make the 5153's aspect ratio different than the 5:6 they chose for BASIC, and then never really told anybody about it (and changed their mind again when they settled on the 5:6 ratio for displaying CGA doublescanned on VGA), or:
#2: When designing CGA IBM just put the number of pixels on the screen that would easily fit into 16k of memory and was an even multiple of their 8x8 character dimensions without really thinking about aspect ratios at all, and the 5:6 number was arrived at by Microsoft as an easy-to-compute approximation that looked fine on your average composite monitor. Later, when the 5153's plastics were being designed, the guy doing that job just made the hole a shape he thought looked good and called it a day. Since WYSIWYG wasn't really a "thing" back then nobody really cared that much that a CGA monitor adjusted to the letter of IBM's instructions (turn the knob until the black outside the borders goes away) made circles a wee bit egg shaped.
Still thinking #2 is substantially (overwhelmingly) more likely, but maybe there was some method to their madness of making that border a little squat for the PAR they settled on later. But I do think I've convinced myself that I ever have a 5150+5153 again I'll go for the stretched borders.