Is "rub on" a phrasal verb or not?

This is a question that well illustrates the inconsistency of analyses surrounding these whotsits.

From UsingEnglish.com

Phrasal Verb: Put on {Separable (optional [except with pronouns])}

Meaning: Start wearing

Example: I PUT my coat ON [You should put some clothes on.] [Put it on.]

(choosing the obviously corresponding usage from those given)

But there is no corresponding entry for 'rub on'.

I'd say that there is exactly the same degree of cohesiveness (unitariness) between rub and on in say 'Rub some lotion on' as there is between put and on in 'Put some clothes on'. (Some would class these particles as 'intransitive prepositions', perhaps to dodge the 'MWV or not?' issue.)

An argument that they should be considered as MWV (multi-word verb) + object constructions rather than simplex verb + PP constructions is the availability of:

Put on your coat. *Put on it.

Rub on the lotion. *Rub on it.

Sit on the chair. Sit on it.

Another is the availability of simplex equivalents: 'Don your coat' / 'Apply the lotion' / *'Asseat the chair'.

However, there are grey areas in the {MWV + DO} vs {V + PP} classification debate. This one is one of the trickier ones.

.................

'whotsits'? I know of at least 3 conflicting usages of 'phrasal verb'. The essential element is: 'Are rub on / take care of etc cohesive enough to be regarded as single lexemes?'