How do you differentiate "soft" CC from "hard" CC?

Each game develops its own terminology.
And even within each game, different people have different definitions of said terminology.
More specific term(s) would/could be: stun, root, snare, mez (and many many more)

But in general.
"Hard CC"
Complete loss of control over your character. This means you cannot move, cast spells or attack in any way for the duration of the stun. In some cases, your characters gets moved under control of another player (pulled, pushed)

"Soft CC"
Any form of debuff that restricts your movement (slow or root) and/or* restricts your attacks (unable to casts spells or heal or auto attack).

• A root and a restriction on attacking would ofcourse result in a hard CC.

I've most often seen "hard" and "soft" CC referring to whether the ability prevents movement or not. Soft CC is usually implied to be any type of CC that still allows a champion to move around. As you mentioned, slows are the most common type of soft CC, since they reduce the movement speed of enemy champions but still allow them to get away using movement skills (or by getting lucky and being able to outrun the chaser, despite the slow).

Hard CC stops movement completely (or forces movement in a specific way). Stuns are the obvious example of hard CC, but this category also includes knockups/displacement (which cannot be cleansed or QSS'd, by the way), snares, and roots. Many assassin type champions are directly countered by hard CC - Master Yi, for example, relies entirely on getting close to his target and auto-attacking them, which he can't do if he simply can't move.