Okay, I think I understand the difference now.
I just noticed that on the website they only give you an ON reading for 晩, which is primarily used for kanji that stand alone (no compounds), so your explanation makes perfect sense.
And I’ll remember not to refer to kanji as “words” while studying them, because that's mostly why I confused the meanings.
Thanks for clearing that up for me!
EDIT:
And also thanks for telling me the difference between 森 and 林.
Nyororin, are you suggesting that I go ahead and use the "Japanese Demystified" book while studying my kanji to get a good grasp of the kanji usage, instead of learning the kanji by themselves? (I was planning on going up to 500 before I actually studied grammar/vocab, and I know about 100 or so now)