The stagger talents do indeed give temporary health (thp) on pushes.
In my opinion, the thp on stagger talent synergizes better with shields than any weapon in the game, since shields have really good stagger. Especially if you like to push mobs to the floor, which the exact playstyle that talent is designed for.
Also, the talent applies to every stagger, but it also is affected by the strength of the stagger. So if you knock 6 enemies to the floor, you will get maximum thp x6. If you just knock them back a little bit you will get partial thp x6. If you knock only half of the enemies to the floor, you will get maximum x3 and partial x3.
Thp on kill is better for weapons like the 1h axe that are always killing things. Thp on stagger is better for weapons that knock enemies around alot like shields. Weapons which do a little bit of both like the 2h axe are really up to personal preference imo.
Hope this helps. Let me know if the way I explained it was overly confusing.