There is a difference in language registers, amoindrir is literary, almost never used in the conversation and rare in books, diminuer and réduire are less expected in a colloquial situation while baisser is not linked to any register.
I'm close to your guesses when comparing these verbs:
Baisser from bas (low, bottom) (lower)
Diminuer relates to diminution, mineur, mini. (smaller)
Réduire relates to réduction (smaller)
Amoindrir relates to moindre and moins. (lesser)
Usage is sometimes different, especially with baisser which is often used in phrases where the other ones can't be :
Je baisse la tête1, la voix, le ton, les yeux, les bras, la visière, le pont-levis...
Sometimes, the three first verbs are interchangeable:
Je baisse/diminue/réduit le prix/le débit/le niveau/les gaz/le volume: all are fine.
Réduire has some specific meanings (shared with the English "to reduce") like réduire une équation (mathematics) or réduire une fracture (surgery).
Amoindrir is rarely conjugated, many of the few occurrences use the infinitive:
Amoindrir le rôle
Amoindrir le pouvoir
Amoindrir le sentiment
Amoindrir le choc
1Je diminue and even more je réduis la tête are possible, but with a huge difference in meaning ;-)