I know this appears to be too basic for this site...
Your main question seems to hinge on the usage of a particle in a phrasal verb, and how to determine if it is an adverb or preposition.
If it looks like a duck, but doesn't quack like duck, it ain't a duck.
A preposition requires an object; if it does not have an object then it is something else, probably an adverb.
Two examples were provided, both using "on" in examples of parts of speech.
rely on it
put it on
Rely on Sth/Sb
In this case, "on" is a preposition which can use almost any type of object.
You cannot say "*Rely it on me" because the object belongs to the preposition. It is not a separable phrasal verb. I would consider this to be an transitive verb + preposition combination, and possibly not a phrasal verb at all. A similar structure is seen in the sayings...
Lean on me,
Depend on our leader.
put on Sth/Sb
...has several meanings, and the idiomatic intent can change depending on the object and its placement.
Put on your clothes/Put them on
...is an example of a separable phrasal verb that can take an object either before or after the phrase. The object of the verb is "your clothes"...but the object can also appear after the preposition because the collocation has an idiomatic meaning. The object of the preposition in this case is unstated but understood as "your body", but other objects might include "a show", or a "song" and have different meanings.
And as Hotlicks has pointed out,
Put Sb on
is a very idiomatic usage that requires the object always between the two and seems to defy the rules...so go figure.