Tomomi plays the Flea style. That is: bass haging low, slapping with the thum pointing down and a lot of bounce. It's the most basic way of slapping, the more natural. Flea (which I love) has a very developed PLAYING, but a very raw technique.
If you're learning slap from a bass professor (and based on your description of the slap and pop technique), it's very possible that he it's teaching you a Wooten/Miller oriented style. That means: bass haging higher, no bounce, thumb parallel to the string (or slightly pointing up).
These differences may be the reason why your're confused. There is no "right way" of playing. Both styles are valid. You gotta find which fits you music better. Developing an advanced technique it's always good, and it will allow you to play faster and more complex things, if that's your goal.
I like Timo's slap. It doesn't sound strange to me. It may not be the most classic slap sound, but I guess that is because she doesn't use a very "slappy sound", like flea.