I'd call the first one Gmaj7 add 13. The intervals are 1-7-10(3)-10(3)-13(6). No 5, no 9, no 11. Because of that, it's potentially confusing to the reader to call it a maj13th chord IMO; it's just missing too much the way you play it. Do you want the reader looking at it and trying to figure out how to put a 5, 9, or 11 in (which you don't play)? That is how most people figuring it out would play it if they saw Gmaj13; they'd try to put a 9 or an 11 in there, and it would sound totally different. Changing it to "add 13" lets them know to skip the 9 and 11. If they saw the "add 13," most would probably play it just like you, except that instead of having the two B notes, they'd fret the 5 on the 3rd fret of the B string, out of instinct to 1) have a 5, and 2) not have two identical notes. The added 5th on the B string woudl not be as you play it, but it would sound closer than what someone would come up with if you wrote Gmaj13 on a chord chart. I'm not sure what the chord notation is for leaving out the 5th, or if there even is one. Stuff like this is why standard notation along with the shorthand helps.
I'd call the second one a Bm7, though out of context, I could be wrong as to what the root is. It goes 1-b7-b3.
The reason you're having trouble naming these chords is the lack of a 5th in either of them. The root and the fifth form a highly discernible "unit," "anchor," "framework," or whatever you want to call it. They provide a glaringly obvious reference point in the audible "blueprint" of the chord. A chord without one of them sounds very ambiguous, and feels as if it is just a transitional chord.
