Thanks to everyone who helped on this - I believe my video issues are solved, and now I'm on to trying to get my touchpad working (I'll start a new thread for this to help with searchability). The suggestions above got both Modesetting and Intel drivers working under Xorg; the reason I was getting the black screen was because my onboot.lst didn't include flwm_topside, so there was no window manager (which is why I wasn't seeing any errors!)
I don't see a way to update my original post to say SOLVED with an explanation (someone tell me if there's actually a way to do this). But for anyone else who runs into similar issues in the future, this was the final resolution.
- I was unable to use Xvesa on my HP laptop with Intel HD Graphics 620 at all. Standard Tiny Core did not work, and in fact I couldn't even boot to a graphical terminal in either MBR or UEFI boot modes.
- Moving to Core Pure 64 instead fixed this. I was then able to boot in both MBR and UEFI boot modes.
- Xvesa is unfortunately not available in 64-bit (several forum posts about this), so Xfbdev and Xorg are the only available options. Xfbdev uses significantly less memory than Xorg, but still not as little as Xvesa (looks like around 118M vs 70M or so)
- After switching to Core Pure 64, simply using tce-load to install flwm_topside and aterm allowed me to start X using either Xfbdev or Xorg (of course I needed to install either Xfbdev or Xorg-7.7 as well, respectively). No other packages were necessary.
- Installing firmware-i915, graphics-KERNEL, and xf86-video-intel got rid of all the errors in the Xorg logs, although flwm worked fine even without loading them. I suspect that the errors had to do with acceleration and would impact performance, but not basic functionality.
Additionally, Curaga was correct - my xf86-video-intel download was corrupt. Deleting and redownloading got rid of that error.