The reason this is important is that not understanding it leads to a lot of confusion and wasted time trying to gain understanding and it always comes back to this:

Dick, you're basically right in your understanding except for the idea that pica and elite refer to typestyles or typefaces. They do not. They refer to pitches only, as somebody said earlier. Now it is true that very many typeface names have the words Elite and Pica in the names but that is does not make them "a pica typeface". You do not have "three elite typefaces"; you might have three typefaces with the word "Elite" in their names, or you might have three typewriters with 12 cpi spacing (which is what "elite" means).

You are already recognizing that Pica Thisname looks different from Pica Thatname and Anothername Pica. That's because "Pica" in a name is only a designator, not a characteristic. It's like expecting Jamie Lee Curtis and Jamie Foxx to look like each other.

The lowercase "pica" is a characteristic: 10 characters per inch horizontal spacing. (Incidentally, it has nothing whatever to do with the height of the characters; it is not like 10 point type vs 12 point type in the typography world.)

I agree with you on "Gothic", too. Gothic originally (19th Century or earlier) meant "sans serif" but it has come to mean also that style which we call Olde English or Fraktur. It's just a quirk of language that the one word designates such radically different styles.

You are also right about the use of all caps being harder to read in big doses. I was not trying to justify your large-and-small-caps typewriter earlier---only trying to give some reason for its existence that wasn't apparent.

So, the upshot is: Just don't think of pica and elite as typestyles and don't pay much attention to their willy-nilly use  in names.

HTH