Is there a correlation between the number of modes in the fiber and the chosen window? In other words, why is third window chosen for single mode fibers and the other two fro multi mode fibers?
This is really an engineering question, because the answer is "cost".
Transmitters for the first window can be made from the GaAs-AlGaAs material system, and they cost substantially less than devices made for the second and third windows.
When using multimode fiber, the inter-modal dispersion is generally limits the bandwidth-distance product of the link, so we don't worry too much about the added material dispersion and attenuation caused by working in the first window.
1310-nm systems with multi-mode fiber have only recently started to gain traction. Since 1310-nm transmitters use the same InGaAsP material system as 1550 nm transmitters, I'm not entirely sure why 1310 nm is preferred over 1550. I have two guesses:
Historically, the 1310 nm window had lower dispersion but higher attenuation than the 1550 nm window. Modern single-mode fiber has optimized the 1550 nm window so this is no longer true. But the available multi-mode fiber still has lower dispersion in the 1310 nm window, and this may be preferred for short-distance links.
A 1310 nm laser will have a lower divergence angle at its output than a 1550 nm laser with the same aperture size. This may make the optics for coupling the 1310-nm laser into the fiber lower cost than the optics for a 1550-nm laser.