Protect your wall tiles with cardboard and take a hammer to the row of counter top tiles that are against the wall. Break them up good. Wearing gloves and using putty knives and screwdrives pull out all the little pieces of tile from that row closest to the wall. Wiggle them out from under the backsplash tile, they should come out fairly easily. This clears a distance the thickness of the tile for the counter to pivot up upon removal before it contacts and loads your backsplash tiles. If that proves to not be enough set the depth right on a circular saw or rotozip and zip through the cement board and pry that off too. It will take more finesse then normally is used when demoing kitchens but with a little caution, it should be very doable.

BEAUTIFUL! You have to enjoy working in your garden, because it is really well tended and planned also. Love it. I see your high chain link fence is like my back fence....7 foot tall and 100 feet long. I've planted Confederate jasmine...or trachelospermum jasminoides....or something sounds like that? It is evergreen here, and makes a gorgeous privacy fence withing a few years. Maybe in 4 years but by 5 it is very effective....they use it in Italy everywhere, including to climb over arbors, up old stairways, and covering fences to create a living privacy fence. Let me recommend it to you for that purpose. If you like, I'll create a Flickr album about the use of this jasmine in Italian gardens (the ones I saw anyway) in Umbria. Etc. Here the cabana was covered in blooming jasmine, the short fence covered in the jasmine is kept trimmed as a short hedge, no hint of the chainlink frame holding it up inside.

In preparing our house for sale, we had our laminate countertop replaced with 2 cm granite. Our installer was concerned about the backsplash and wanted to make sure we had extra in hand in case any cracked. He did have to grind down some of the base tiles, but all went smoothly. By the way, the backsplash is 4x4 Italian porcelain tile (I don't know if that makes a difference).

There very well may be gaps that the splash is covering that the tile won't, however, you'll probably be able to float your wall to make it work. The differing height tops won't look odd with tile. Place a continuous shim the thickness of your grout lines on your main top to start tiling. Install the tile, grout, pull the shim, then fill the gap with color matching caulk.