I was bored so I re-made the badges' design proposed by Jeff Schaller. My take on it is according to the aviation.SE, which has the biggest badges that I managed to find on SE (increased here by a factor of 10):
I draw the black pixels to show how I measured it, the image suggests that SO/SE badges are 16x16 pixels sprites. The aviation.SE badges are not round, therefore I believe that not round badges are OK.
Anyhow, for U&L we could try this (increased here by a factor of 3):
In the actual badge size (16x16 each) these look as follows:
Used the SE colors, as per comments:
Also, created a white version:
And a very white versions (not sure which one scales better):
(The images are 16x48/16x56 but the badges are actually slightly smaller than 16x16, to fit better in the 16x16 container)
The font used is Liberation Mono, a version that is already under the SIL open font licence. Therefore no copyright problems exist.