Let me give you a slightly silly example of how I see Te vs. Ti:
When using a computer, some people have a real "need-to-know basis" mentality. They'll either memorise or write down the precise steps to achieve each individual task. This is how Te works, IMHO: sequential and contextual in its creation of rules.
Other people won't be satisfied with a list of "if-then" cases -- they want to get to the heart of it in order to liberate themselves from particular contexts. In other words, they want to figure out the underlying principles apart from any specific cases in order to be able to act outside of such a restrictive framework. This is Ti.
So Te will be satisfied with "If you want to do wordprocessing, click the Word icon", but Ti will want a deeper understanding: "Programs can be run by clicking their icons (which are themselves pointers to the actual executable files)".
Like I said, it's a silly example, but it kind of paints a picture: Te is tied to specifics in its creation of rules, whereas Ti generalises principles free from any specifics.
I'll have to ponder your Ni/Si ideas further. Ni, to me, is a kind of tapping into the underlying archetypes in a very focused way. (David Lynch has a very interesting way of describing this, actually.) But this is all for another thread.