Here's my take. You wrote here idiomatic phrases and expressions(ex. needless to say is not “random”), which do not necessarily equal idioms but they have an idiomatic, common usage touch. These are also called phrasal idioms, idiomatic/common expressions, set phrase, etc. And they have subtle meanings, given by you, which however can be obscured in common language.

In addition to that, phrase and expression have additional strictly grammatical meanings. So that a phrase consists of typical two or more words (what Kate Gregory mentions). A phrase can be a sentence but usually is not. An expression is the more general case and can be a word, phrase or sentence

So at the end, actually "idiom" has the clearest definition (established meaning not directly linked to the individual words).

First, I want to thank you @Bogdan for emphasizing this question. I tried to do what others did, and classify the three terms using a transitive relationship. Not quite that simple. The more I thought about this, the closer I got to erasing my entire answer and giving up! In response to your fine comments: I did not say that phrases (such as "needless to say") were random combinations of words, not semantically linked. But I remember reading that on ESL yesterday, yet can't find it now! Kate Gregory conveyed better what I meant re categorizing phrase vs expression as sentence or fragment.

@Bogdan con't I will clarify that part of my answer accordingly. Also the grammatical definition via your citation is sensible, I will incorporate that too.

Please don't erase it-note that this question had a bounty from me and you got it. I didn't say your phrase was random, on the contrary, I wanted to emphasize that it is not. So from your comment I see that you wanted to say the same thing about phrase as Kate Gregory(I misunderstood before that you diverged on that point). So now I see what you meant by semantical (which is the same as what I meant by grammatical). But note phrase could in addition mean idiomatic phrase. So there are two meanings of "phrase" and "expression".

@Bogdan Thanks again for clarifying. I was wondering if I should alter an answer which was bequeathed a bounty (sorry, but I'm a relative novice here, so am very happy and excited). My knowledge of grammar is not as good as it could be. Thus some terms like phrasal idioms are unfamiliar to me. Studying Latin (or recalling French grammar better than I do) is almost de rigeur <grin> for being able to articulate grammatical constructs. This last comment is off topic, I will delete it after a brief while, after I think you have seen it. <Idle thought: I wonder if there is IM or IRC here?>