Alex,
First, you did the right thing by removing the old BB if you expect long life out of your bike rebuild moving forward.
If the crankset and chainrings are not worn, then replace the entire BB with the UN55. Be sure to get the Shimano version and not an aftermarket BB...the last link you show is correct. Spindle length is critical for the best chainline...so get the 113mm spindle length if that is what your old BB is either marked or measures.

Many in your shoes would replace the entire crankset with Ultegra with the gearing you may prefer...such as 50/34. Modern cranksets are much better and less problematic than old square taper but many on this forum still own square taper BB bikes and like them just fine.

Good luck with your build.