5:52 PM
Have you tried just to write a computer program to find the exceptional values? They are indeed, finitely many; as soon as the region contains a unit square in every direction (for which it is enough to contain a disk of diameter $\sqrt 2$), it has an integer point in it regardless of the orientation, so for all $b$ with $|b|>15$, say, all $4$ possibilities are realized and (taking symmetries into account), you need to check just about $100$ values, which should be a piece of cake. — fedja 22 hours ago
@fedja I have posted an answer showing the distribution for small enough numbers. I am not sure to which extent it is similar to what you intended.
I have to admit that I thought that there would be more exceptions. For some reason I kept thinking about integer multiples of $\frac1b$ and integer multiples of $\frac ib$; but somehow I ignored that also their sums come into play.

6:17 PM
The pictures are cool :-) What do you use for making them? (I usually stick with Asymptote for most of my drawings but I'm curious what the preferences of other people are in this respect)

I am pretty lame in drawing pictures - MetaPost is basically the only thing I am familiar with well enough to create some reasonable picture.
WP: MetaPost Here is some kind of manual: tug.org/docs/metapost/mpman.pdf
I'd guess several other tools are more popular:
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I create the diagrams for my (La)TeX documents using ... Possible choices: TikZ, PGF, PSTricks, xypic, picture environment, MS Visio, Inkscape, ... Select 'PGF' if you really using the low level PGF commands, but 'TikZ' if you use that higher level interface.

From MetaPost I get the picture basically in ps (or eps) format. Which is probably good for use in LaTeX documents, but for use on Stack Exchange I have to convert them after that.