Um, you're mostly on-point there. Some quick things to review:
Here's a short list of PS3s that you can play disc-based PS2 games on:
- 20GB Launch Model (hardware based PS2 emulation, no Wi-Fi)
- 60GB Launch Model (hardware based PS2 emulation, Wi-Fi)
- 80GB 1st hardware revision (software based PS2 emulation, there are a few flavors of this model)
All PS3s play disc-based PS1 games.
Launch model and early-gen "fat" PS3s can be prone to thermal breakdown (the yellow light of death), so if you buy a used "fat" PS3 keep that in mind. It's not as widespread as the hardware flaws found in early gen 360s, but it's a noted hardware issue. Sony does repairs on all makes/models for around $100/$150.