She does wake me up in the night sometimes, though.

That's her? Gorgeous!
Mine is more of the traditional sort - more round headed, though not really an "apple head". He's a hoot, for sure. And yeah, it does seem like they're telling you about their day and what they think about things! When I first got Tam, he drove me nuts...because I was constantly trying to figure out what he wanted. After awhile I realized he didn't necessarily want anything - he was just talking.
He has a few different "words" that I recognize now, though. Like he has a specific yowl that means "Halvah, come here!" He doesn't use it for anything else.
I'm so glad you've discovered Siamese too. Everyone should have such a loving pet. They've made my life so much happier since my friends move into my place with them. I re-discovered my love of animals from my childhood again, a part of me I had forgotten over the years as college, career-related moves, etc. all took me into a transient lifestyle. Only now having been in the same place for 7 years with my friends moving in have I rediscovered that neglected part of me again.
Like you, I've found they just love conversing with you, even if either side doesn't really understand what the other is saying. And they're always doing things that make us laugh. They're real comedians.
Hey, whatcha going; why are you sleepin? Nighttime is the best time to hunt and other stuff... come on get up get out of that bed...
I have a half Siamese/half only God knows what that I got from the SPCA. Her face looks exactly like your Siamese. But her body is partly white with Siamese colors interspersed. She has many of the Siamese traits such as following my son around WHEREVER he goes. And her fur is incredibly soft. Her only bad habit is sleeping at the top of stairs (ANY STAIRS) and refusing to move so you can get past her.
My, my, my what a beautiful girl! Just love her smarty-girl eyes. Our Taylor could be her twin.
Nice face, that one is a purebred stray. Lucky you.