Idioms, phrases, expressions and sayings

I use the category label of "idioms and phrases" on this site, but I sometimes rather cavalierly call multiple-word phrases "sayings," "expressions," or just plain old "terms." Are they truly interchangeable? I decided to investigate this confusion. 

An IDIOM is defined as a group of words established by usage as having a meaning not deducible from those of the individual words. "Raining cats and dogs" and "see the light" are examples. 

A PHRASE is defined as a group of words that works together in a sentence but does not contain a subject or a verb. Often phrases are used to describe people, things, or events. The English teacher in me notes that I used to have to teach these phrases in grammar units: Noun Phrase; Friday became a cool, wet afternoon. Verb Phrase; Mary might have been waiting outside for you. Gerund Phrase; Eating ice cream on a hot day can be a good way to cool off. Infinitive Phrase; She helped to build the roof. Prepositional Phrase; In the kitchen, you will find my mom.

But when I label something as a phrase on this website I don't mean those things. I am more likely talking about simple expressions, idiomatic expressions, and the "turn of phrase." Still, I find other websites that also lump idioms and phrases together.

There are plenty of these that don't need an explanation or origin story from me: "A hard nut to crack" meaning a person who is very difficult to deal with is obvious.

What label would you put on "A picture is worth a thousand words" or "Better late than never" or "Born with a silver spoon in mouth"? 

I can say that idiom comes from late 16th-century French idiome, or via late Latin from Greek idiōma oddly meaning "private property" from idiousthai  "to make one's own" which in turn comes from idios meaning own or private. Is suppose the idea was that these phrases began as some personal expression and then some became more widely used.

Okay, I suppose I keep using the idioms and phrases" category and also use the other terms.