A little late on the reply, but here'* what I know (I am familiar with these cars/drivetrains, but by no means do I know it all).
As from what I've heard/read the 4T60E was available until 1997 in the non-supercharged 3800, 3.1, 3100, and who know'* what all else.
The 4T60E-HD was available 1992-1996 on the supercharged 3800 (and maybe the 3.4 DOHC in GTP and Z34 and so equipped Cutlass Supreme'*). This HD version of the 4T60E was standard, and the ONLY transmission used with the supercharged 3800.
In 1997 the 4T65E-HD replaced the 4T60E HD in supercharged applications, but in the non-supercharged 3800'* the 4T60E non-HD remained, although the Park Avenue and Riviera did get the 4T65E non-HD a year before other L36 (non supercharged Series II 3800'*).
You are correct about the ID #'* of the trannies. One difference though is although they are both 4-speed (forward) transmissions the 4T60E is a 3 spd, with the 4th gear being an overdrive (notice the gear selection difference between your 1996 and your brothers 1997), and the 4T65E is a true 4-spd without OD.
As for Gonneville'* info he is correct, if you have an L36 - the 1997+ L67 (Series II supercharged 3800) only had the 4T65E-HD. To put an HD tranny in a L36 car would require PCM work to program for the HD gearing (which IIRC is 2.93 and the 4T65E non-HD was available with 2.84, 3.05, and 3.29 gearing -3.29'* being available only in Grand Prix GT'* and maybe Regal LS'*).
I forgot about this site when listing tuner'*, but for your 1996 you might be able to find someone to reprogram your PCM at www.L67swap.com
as there are a few people who have swapped SC'ed 3800'* into other cars, namely Fiero'*