I speak Chinese :)

It's a mixed system with about 2 millennia of legacy. It started as logographic, then it got into phono-semantic compounds, with detours into the written-only official language (like Latin), and now it's messy mix of everything. There are true logographs (休，林，森), true phonosemantic compounds, and plenty purely phonetic characters that have no meaning by themselves ("bound morphemes").

> now it's messy mix of everything. There are true logographs

Don't confuse the origin of the system with what the system is now.

Using your example, what do you see as the difference between the "logographs" 森 and 林?

Neither can be a logograph, because neither one represents a word. But even if that weren't the case, on the assumption that they are simply pictures representing concepts, how would you know which one was which?

What does it mean, to you, that the word "forest" must be written 森林 and not 森?

> and [there are] plenty purely phonetic characters that have no meaning by themselves ("bound morphemes").

...yes. 森 and 林 both belong to that category. But you've specifically contrasted them with it. I can't tell what you're thinking of.

Characters can be classified by origin, so that 森 is "从林从木", 切 is "从刀七声", and 下 is "指事". You seem to be reaching for this, but "bound morpheme" is a classification of the current use of the linguistic element, not of the origin of the way it's spelled.