Yes, square wave is the worst, not only does it cause the fan speed to drop considerably, it also causes tube lights to glow in the dark. It is also harmful for your electronics. Modified sine wave will definitely slow down the rpm of motors as motors (both inductive and/or capacitive) are designed to work only with pure sine wave. For anything that has a motor built in like computer hard drive, electric machines, water motors, fans etc only pure sine wave is recommended. Also sometimes printers do not work well on modified sine wave and print garbage on the paper.
Anyway, back to the problem at hand. The UPS does have overload protection and it is not the issue of overload, as I only had one fan running on it. There might be a problem due to dimmer as you mentioned as the fan in the other room does not slow down that much, I checked today. I will have the dimmer in my room checked or replaced and see if this solves my problem.
I found the Homage manual and the narrow/wide is the input line sensitivity control. In narrow mode the UPS operates within 170-280 VAC input with 230 VAC output (transfer time 15ms). In wide mode it operates within 90-280 VAC input with 230 VAC output (transfer time 48ms). This setting is designed to accommodate voltage fluctuation on the input line. Wide is recommended for Pakistan because we have the occasional voltage fluctuation due to unreliable service.
Are you using Homage inverter as well?