OK, it sounds like you have a plan.
If you get bare LED strips, most run on 12V DC. If you want to power it from the Accessory voltage on a transformer (14V-16V AC), there are AC-to-DC converter modules on eBay (or elsewhere) for about $3.
The point about foam board was if you wanted a uniform, diffused back-light on the rink surface rather than hot-spots. So as shown in the video, if placed about 1" or so above the LED strips, you can't make out the individual LEDs. In general, individual LEDs have a narrower, conical light output whereas the LED strips have a broader, wider beam (depends on what you're after of course).
In case you're not familiar, I show a Red-Green-Blue LED strip along with a RGB LED controller module at the bottom. The module has 3 flat button right on it to program it to do dozens of different things - sweep thru all the colors, adjust brightness, display only a single color, flash, etc. Or for another dollar or two you can get a fob remote to change the lighting pattern.
I realize I'm over-complicating this for your project but maybe it can help with another scenery project.