Tile – Cement Board vs Dry Wall

Unless you've got a lot of intricate cuts (tough with concrete board) or expect to have a LOT of water on your backsplash on a regular basis, it's probably just simpler to go with backerboard.

I have used a Kerdi-board competitor, Wedi, for a shower and found it very easy to work with. Easy to cut and seal. Comes with sealant (caulk tubes) and fastening bolts (might be tricky to drive in small backsplash places) To avoid flex, you want to use a thicker board. It's pretty strong stuff (particularly for a kitchen backsplash, which shouldn't bear much (any?) weight or pressure.

I have also used the original Kerdi membrane (not the newer board you mention) for a shower. I have a lot of confidence in the quality of the seal with the Kerdi membrane, but it's quite a bit of extra work if you haven't done it before, and is probably overkill for a backsplash (I assume the counter top runs below).

As for cost - my personal opinion is that small jobs are precisely the place to spend the extra money (assuming the quality is better for what you need). If I had to do 100 bathrooms, the added cost of the these newer products would really add up. If I'm doing 1 or 2, then the difference is relatively small.

As a bit of history: The Kerdi membrane (from Schluter Systems) is one of (the?) original products in this space. Wedi board came as a competitive product, eliminating the need to put up drywall and then apply the membrane. Schluter responded with the Kerdi board.

You just want a stable platform for your tile. It depends on the size of the tile.. If you are cantilevering 10-15% of a larger tile, just fill the thinner areas with more mud and put backer where it will fit.

It looks like you needed some blocking behind the green board (where the rear curve comes in) to be able to fill in. Perhaps some fiberglass mesh tape would let you land some thinset to bridge the gap.