I believe it's bartonella that causes red eyes in Lyme and coinfections. After I started the bartonella treatment, I have not had red eyes since. I remember waking up in the morning and they would be almost completely bloodshot. Bartonella is more an infection of the capilliaries in the eyes.
However I do think Lyme infects the eyes as well, but just doesn't cause the red blood shot eyes. Instead it just drills and bores it's way around tissue trying to get to the collagenous areas. So it likes to get into the fluid of the eyes.
One thing I used to look at the white snow in the bright sunlight and I could see spirochetes floating around in my eyes. They are long super thin, barely noticeable but definitely there. Now after doing the tinidazole I can't see any spirochetes floating around in my eyes anymore from the bright light, so I think the medications are working. For awhile there they were just reducing in number, now there might only be the rare occasional one, rather than a whole bunch.
So I think both Lyme and Bartonella, both infect the eyes, just in different ways. Bartonella infects more the capillary, causing the red blood shot look, whereas Lyme actually goes into the viseral fluids themselves.