What's the most accurate term for phrases such as "storm in a teacup" and "making mountains out of molehills"?

Those are generally called idioms.

Many idioms, including those you used as examples, are also metaphors, which you can learn about by clicking on the link. However, not all metaphors are idioms, so metaphor would not be a good word to describe all such sayings.

Expression and saying are also used of such idioms, but, as you pointed out, those terms are more general - too general, in fact.

Proverb is also not an appropriate term to use for these idioms. A proverb is usually a complete sentence, and it always expresses general wisdom on a situation. It is not a term, as an idiom is.

Anecdote is probably the furthest off from what you want here. It basically means "a short story or historical account".

These are clichés, and they are best avoided except for ironic effect.

They are also idioms, which might be the word you're looking for.

I prefer to think of them as tired and hollow, since their meaning and certainly their impact has been long since beaten out of them through years of abuse.