^ I think it's incredibly oversimplified to think that Descartes argument compared to Nietzsche's is a perfect analogy to Ti and Te. There is always more than just Thinking at play in any argument. It takes N (which takes S) and F to come up with the premises; all Thinking does is analyze the structure of the argument in a binary way.
The definitional difference between the Thinking functions (which aren't even mutually exclusive) is that one is focused on the external standard and one is focused on the internal. In other words, Ti uses the metric relevant to the current internal thought process to judge the correctness of something, and Te uses the metric relevant to the current external situation. Both functions have only two possible outputs, though (true or false, 1 or 0, whatever you want to call it). Ti outputs true or false as a function of whether or not something fits in the thought process, Te outputs true or false as a function of whether or not something fits in the environment. So I don't really think soundness and validity perfectly describe their relationship.
As computer functions, they'd look like this (I'm just making up a language here):
Te (takes two inputs, e (the current environment), and x (something to be analyzed)):
If (x is consistent with e)
Ti (takes two inputs, i (the current internal thought process, conscious and unconscious), and x (something to be analyzed)):
If (x is consistent with i)
There are actually plenty of times that i and e aren't very different -- a given x would cause the same output using either Te or Ti.
I've gotten sort of off point, but I really just meant you can't think of Te and Ti as such different types of functions.
I think in general Ti is more concerned with internal consistency for this reason -- the internal standard itself changes with every thought -- once you start stacking premises, the internal standard will hold on to all of them and check them against each other (because the premises actually become the internal standard, as the user is currently thinking about them).
Te doesn't care about that kind of thing, because it just stays focused on what is environmentally relevant. When you feed Te a list of premises, all it does is check each of them against the environment -- the standard Te uses does not change as much with each input. If any premise contradicts the environment, Te just say "false" and be done with it, the function itself is not interested in whether it is hypothetically true given some other premises not visible in the environment.
That's why I like to simplify the whole thing by saying:
Ti - true/false
Te - works/doesn't work
I think that catches the actual mechanisms at play.