You're tripping up on terminology, which is understandable since it's hard to find reliable information about English grammar, especially online. Everybody uses their own terms, with whatever meanings they think those terms should have. Let's see if we can clear some of it up.
First, "phrase" and "clause". Unless it's named (like noun phrase, verb phrase, prepositional phrase), phrase just means some connected chunk of language, which should be a constituent of a sentence.
Clause, on the other hand, when referring to English grammar, means a structure composed of a verb phrase, along with a subject noun phrase. All sentences contain at least one clause, the main clause; most contain more clauses. The subject noun phrase of a clause must be present in the clause if it is a tensed clause -- i.e, if the verb in the verb phrase is present or past tense; that-clauses and embedded question clauses
- I suspect [that he is already here].
- I wonder [when he arrived].
are the two varieties of tensed clauses.
But gerund clauses are not tensed; they're "non-finite" clauses, with neither present nor past tense, but rather a special form of the verb marking the construction.
This -ing form of the verb is unique, but has many uses. It's used in the progressive construction, with an auxiliary be; that's not a gerund. It's used to modify nouns, like the man standing in the corner, but that's not a gerund either. As you point out, it's when they get used as subject or object (or prepositional object -- some nouny use) that an -ing clause is considered a gerund. Many don't use that term, but it's safe, as long as you're talking about subjects or objects in a higher clause.
So, if a gerund is a clause, where's the subject? Well, subjects are not always required for untensed clauses like they are for tensed clauses and main clauses.
Instead, untensed ("non-finite") clauses like infinitives and gerunds have Complementizers -- special phrases or grammatical constructions that introduce the clause and mark it as being one special kind of subordinate clause. Tensed clauses have complementizers too, like that or Wh-words, but they're more important for non-finite clauses.
Infinitive clauses have a complementizer called for...to. It consists of two prepositions; for can introduce the subject noun phrase and to can introduce the verb phrase. For is usually missing, except at the beginning of a sentence, and is always missing if the subject is. To is usually present, but there are some verbs that can take infinitives without to.
- [For him to leave early] would be a good idea.
- She said [for him to leave early].
- She wants [him to leave early].
- She saw [him leave at 3.]
Note that the subject of an infinitive, when it's present, is either a noun, or a pronoun in the objective case -- him, me, her, us, them.
Gerunds are like infinitives in many respects - they must have a subject, though it might have been left out; it can still be determined, however. And the subject may be in EITHER the possessive (my, your, his, her, our, their) OR the objective case. The possessive variety (his leaving early) is called the POSS-ing complementizer, and the objective variety (him leaving early) is called the ACC-ing complementizer ("objective" used to be called "accusative" because Latin).
The important thing here is that both are grammatical and there is no difference in meaning between the two gerund complementizers. Either one can be used anywhere. Some people think that using POSS-ing is somehow more formal, elegant, precise, or correct than using ACC-ing. This is nonsense, like other zombie grammar rules.
So, to get back to the OQ, it's not a trick, it's a complementizer, and it's only one of two gerund complementizers. You could say either one and you'd be correct:
- I didn't watch him mixing the cement.
- I didn't watch his mixing the cement.
As to clause/phrase/gerund -- don't use phrase unless you're prepared to say what kind of phrase; otherwise it's handwaving. Think of clauses first; any non-auxiliary verb defines a clause, so if you count the verbs you've counted the clauses, many of which are likely to be reduced. Gerund is just the special name for a complement clause using a gerund complementizer.
As for the final question, that's just your taste. You're not required to say it, but you're not allowed to correct anybody else's usage unless they request you to.