I usually understand pessimism/optimism in music to be an attitude presented by the lyrics of a song; for example, if it shows hope or if it shows despair. It usually always does go hand-in-hand with sadness (for pessimism) or joy (for optimism), but you are right that they are certainly two different things.

Like I said, a lot of it has to do with how you personally interpret the lyrics. I tend to see a lot of raw despair and weakness in many of Ayu's songs, and a lot of acceptance of sorrow. For example, in A Song for xx, the key pessimistic points would (in my opinion) be:

(1) Hito wo shinjiru koto tte itsu ka ura kirare
Hanetsukerareru koto to onaji to omotte ita yo
I thought that trusting people was the same
As being stabbed in the back and rejected.

(2) Hitori kiri de umarete hitori kiri de ikite iku
Kitto sonna mainichi ga atarimae to omotte'ta
I was born alone and I will go on living alone.
I always figured that was just how life went.

The final chorus of M displays both pessimism and optimism. They way Ayu says she sees things is that everyone is unhappy (Pes.), but that she wants to be happy (Opt.). In other words, she is pessimistic, but wants to be optimistic. She also prays that this was is her last love, so that she doesn't have to feel this way again. I think that's a boldly pessimistic request.

Because of You is mostly just an agonizingly heartbroken song, but the one final line makes it pessimistic to me: "It must be [because of] you." There is also one line that is repeated at the end of each chorus (albeit with different verb inflections), which expresses how Ayu almost wishes she had never met the person. The emotion the statement is founded in is sorrow, but the statement itself is pessimism.

In addition to POWDER SNOW, I think the best examples are still HAPPY ENDING and HOPE or PAIN. The irony in HE is pessimistic enough, although the final line is really what brings it about:

Sou ne watashi ni wa koufuku na ketsumatsu nado
Niawanai koto mo dare yori wakatte iru
Yes, I know better than anyone else
That a happy ending doesn't suit me.

There have been lots of times when Ayu has admitted defeat, though, particularly between Duty and MY STORY. She shows a lot of weakness and despair, and feelings of expecting the worst from things. There are sad songs that have hopeful twists (Endless sorrow, NEVER EVER, Daybreak, SEASONS...), and there are songs that are just plain sad without being negative or pessimistic (HANABI, Over...). But there are also a fair shair of songs that don't have much of a silver lining to speak of.