So what is the mentality that makes a person categorize phrasal verbs into, especially, type1 and type2?
The mentality is one that wants to categorize things precisely and state precise, always-dependable rules, of course!
Communication is a complex skill involving empathy, aesthetics, logic, creativity, awareness of common knowledge, and much more. The great majority of this kind of knowledge can never be put into words. But, some people try. Some people hope to speed learners past all the cumbersome trial and error that happens when you learn by doing, and some people want official rules to determine in every case what is correct and what is incorrect. Some people want rules to settle arguments. Some people want detailed scientific understanding of how language works, which requires categorizing things much more finely than you need to for everyday conversation.
All those mentalities have a natural alliance when it comes to inventing more and more precise categories in order to make their rules more and more complete. Doing that can lull a person into thinking that language is just abstract rules and symbols. People can forget that language is a real, biological phenomenon, which works mostly by imitating and varying what you've heard, and trying to speak so that others will know what you're imitating and how you're varying it.
When the categorizing and rule-articulating mentality goes too far, it sometimes leads to an unfortunate way of teaching. You might have encountered it. Instead of cultivating your ability to express yourself in a language, mostly through practice and through exposure to influential things that most native speakers have seen, this way of teaching works by mostly telling you lots of declarative facts and rules and requiring you to recite them. This is unfortunate, because learning to recite rules makes you skillful at reciting rules, but not at communicating. For communicating, you don't need a detailed, precise theory of English grammar. I think learning too much theory might even be damaging before you've learned to express yourself spontaneously and clearly in English.
You do need to clearly understand a few facts about English grammar, though. One of those facts is that English has phrasal verbs. If your native language doesn't have them, and you don't know about this quirk of English, they can be very hard to figure out. But it doesn't matter whether you call the second word a preposition, an adverb, a particle, or a rototiller.* The main thing to know is that English has a lot of special phrases that mean something different than what you get by combining the individual meanings of the words—and in phrasal verbs, the prepositions (or whatever you call them) don't introduce prepositional phrases. (I wrote a little more about this here.)
English grammar is kooky. It's hard to learn because you have to master many phrases that all work in their own peculiar way, without any common, simple rule to explain them all. The way to learn it is to practice using specific little phrases in real or at least realistic situations. After a while, you'll notice patterns among these phrases. And after a while, you'll begin to develop a feel for how to create similar phrases of your own, and anticipate how a fluent speaker will react to them. You don't need to memorize lots of terminology.
After a few years, you might even understand what up means in English. But you'll never be able to explain it.
* Actually, it does matter. In most situations, you should call it a preposition, because that's what most people call it. It's common knowledge that people call it a preposition. Most people have never heard it referred to as a "particle". But you can only know what's common knowledge and what isn't by getting to know the culture, not from reading declarative facts about it.