I want to notice one more time: in russian particle "не-" negates next word and (it is important) other words stay intact. As I said, rules for "not-" in english for me were confusing in some way.
Let's look one more time:
"Не я хочу это" = "(It is not me who) wants it" ("want it" stay intact. sentence has information that something else "want it")
"Я не хочу это" = "I do not want it" ("I ... it" stay intact, but it is the same as in english)
"Я хочу не это" = "I want (something, but) not it" or "I want something else" ("I want" stay intact. sentence has information that there is something else what you want)

This is important, because there is a little difference, for example, between:
"вы бы не так заплакали"
"вы бы так не заплакали"
By the fact it is the same idea. But there is some shift of accents.
"вы бы так не заплакали" = "cry" is negated. so... "if you will have to cry, you will not do it like this"
"вы бы не так заплакали" = "like this" is negated. "if you will have to cry, you will do it in other way"
The meaning is identical, in this case, but these are different phrases. You should notice it. This difference - which word is negated - can be very important.