That is a gorgeous floor. I also like the acid treatment you sometimes see on Houzz. Here's an example:

Yes, the first picture is polished concrete. It looks like it was sealed but not stained. Concrete floors with radiant heat are a winning combination! There are so many options when it comes to concrete floors. Polishing can be great in the right place. With staining the color options are growing. Acid gives a great look but with a limited palette, messy clean up, and some colors contain chromium. There are great new stains and dyes in both water and solvent bases that will offer about two dozen color choices that can be blended into any number of combinations. I'll attach a picture of a sample tile that was created with only water based stains and dyes(over a thin micro topping). It was layered like you would create a water color painting. Concrete micro toppings can cover an existing slab as thin as 1/16" to resurface an ugly slab, add a pattern, change the surface texture, etc.

Thank you so much.