Van Beuren's Tom And Jerry are as far removed from Hanna-Barbera's MGM Tom & Jerry as you could ever find!
I <I>think</I> the VB ones were based on a Vaudeville act of some kind, going by shaky memories here, and literally were a couple of guys that did silly things.
Years later, MGM put out a cartoon called "Puss Gets The Boot". At this time, it featured a mouse with no name and a cat called Jasper. If I remember correctly, a second cartoon also used the name Jasper for the cat. With these becoming successful, MGM asked HB to formulate an official teaming of these two characters and start a series.
The cat was called Tom ("Tom Cat", geddit?) and once that was in place, I would imagine that Jerry either followed because someone remembered something like that from Vaudeville, or it was a direct nod to the earlier series (more likely that, with that series' demise, no-one would remember the VB Tom and Jerry, so a name lift wouldn't have been so noticable).
In any event, the characters never became confused with each other until home video allowed a series of public domain releases to come out, starring "Tom And Jerry". Naturally, some people were annoyed they'd picked up a cheap disc which was nothing like they had expected, while others recognized the vintage and rare nature of the material.
With the "Rainbow Parade" series, Van Beuren actually put out some nice cartoons directed and scored by some later big names. He is not related to the President, as far as I am aware.