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I'm confused about your post title, but these naming conventions are how music works.

Time Signatures

A time signature is written in the form of X/Y where X is the number of beats per bar, and Y is the note length that denotes one beat.

Examples:

4/4 time means there are 4 notes per bar, and a quarter note (1/4) is the length of one beat.
3/4 time means there are 3 notes per bar, and a quarter note (1/4) is the length of one beat.
6/8 time means there are 6 notes per bar, and an eighth node (1/8) is the length of one beat.

Note Duration & Naming Conventions

There are different types of note lengths as you've mentioned and they break down as follows:

Whole note (The longest duration that can be written). These are referred to as Semibreve in Britain.
Half note (1/2 of a whole note). These are referred to as Minim in Britain.
Quarter note (1/4 of a whole note). These are referred to as crochets in Britain.
Eighth note (1/8 of a whole note). These are referred to as quaivers in Britain.
Sixteenth note (1/16 of a whole note). These are referred to as semiquaivers in Britain.
Thirty-second notes (1/32 of a whole note). These are referred to as Demisemiquavers in Britain (lol).

Triplets are a bit of a special case, as they take up 1/3 of a given note type

Quarter note triplets split a half note into 3 equal parts. Quarter note triplets are known as crotchet triplets in Britain.
Eighth note triplets split a quarter note into 3 equal parts (in dreams this is generally what 1/12 is). Eight note triplets are known as quaver triplets in Britain.
Sixteenth note triplets split an eighth note into 3 equal parts (in dreams this is generally what 1/24 is). These are commonly referred to as sextuplets in the US, and as semiquaver triplets in Britain.

Hopefully this helps you understand how the music grid snap is broken down.

My advice is to not change the time signature per instrument (this includes the playback speed) until you have a better grasp on these fundamentals. I feel very comfortable with this knowledge and I've screwed myself over in Dreams by doing that in the past. (E.g. I wanted quintuplets (5 beats per quarter note), and this makes reading the *inside* of an instrument pretty difficult).