Yes, these are the good words here.

Souvenir is for something you remember, a remembrance… It also design an object who recall a moment of your life to you (like an object you got on a trip).

“J’ai souvenir d’un long voyage…” -> I remember a long trip…

“Je me souviens de ce jour…” -> I remember this day…

“J’ai acheté ce souvenir…” -> I bought this token…

Mémoire has approximately the same definition than memory and can be sometime used in place of Souvenir :

“J’ai mémoire d’un long voyage…” -> I remember a long trip…

Yes, memory translates to different terms in french, depending on the meaning.

As you presumed :

A memory of walking in the park (memory = souvenir)

but

I have an excellent memory. (memory = mémoire)

I can’t think of any other meanings…

You have the right word for each context. A single memory (i.e. an event) is a “souvenir”, but memory in general is “mémoire”. Other expressions that have more quirky translations include:

• “in memory of” → “à la mémoire de”/”en souvenir de”
• “have pleasant memories of” → “garder un bon souvenir de”

In computing, “memory” is also “mémoire”: “Mémoire vive” “mémoire morte” (“RAM”/”ROM”).