As noted in the comments, phrasal verb is a slippery term that has been variously defined. Indeed, the Cambridge Grammar of the English Language states (p274): We do not use the term 'phrasal verb' in this grammar.
However, if we follow the definition given in the foreword of the Collins Cobuild Dictionary of Phrasal Verbs, then phrasal verbs are "combinations of verbs with adverbial or prepositional particles".
So the answer to your question: "When is a phrasal verb not followed by a preposition?" is "When it is followed by an adverb(ial)." For example: to steal away, to cut back.
Other phrasal verbs have particles which can in other contexts function either as an adverb or as a preposition: to put down, to get across, to come about.
The last group contains particles that can only function elsewhere as prepositions: to make do with, to get at, to make of.
The question in your title is a complex one. The textbook that you refer to should be able to help you answer it.
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?'
Best Answer
In each case keep is the verb and off is the preposition.
In "keep off the grass" there is a contraction - the object is "implied". It's actually an imperative: "Keep everything off the grass". verb-object-preposition-noun.
In "Keep your hands off her", it's a pretty straightforwards imperative. verb-object-preposition-noun.
You can tell the 'keep off' is not a phrasal verb, because it doesn't have an idomatic meaning and because the preposition off clearly refers to a noun. This is in stark contrast to "break down" (as in "break down and cry"). In the latter case, the phrase has an idiomatic meaning, and the word "down" does not link to anything else, it is intrinsic to the phrase.
In the case of "keep off the grass", "keep" is a verb, which means "continue or cause to continue in a specified condition". The specified condition is "off the grass". Clearly off is related to the grass, and not part of a phrasal verb.