Your backsplash is a dated look and will probably be damaged when you remove your counter so IMo Plan on new and it will be fun to change it too. As for counter I like real stone not plastic quartz . Your cabinets look quitewhite so I would suggest you look at white granites and use the same for the backsplash IMO the easiest and simplest look the little 4" piece with the tile above but no all the way is not a good look If you want tile with the new counter then no 4" pice just tile from the counter up and that tile will be decided after the counter is chosen and even better if you wait until the new counter is installed .Make sure all the lighting is the same I like LEDs 4000K to keep color true all the time

I guess it depends on you budget.

Are you in love with the backsplash? You can get rid of it now fairly easily and then you wouldn't have to decorate around it since that would limit your choices. Even if you didn't replace the backsplash immediately because of budget your options would still be open to what you loved.

Most people choose the counter before the backsplash because there are so many beautiful tile options whereas finding the perfect slab is more difficult. Same reason most people choose paint as one of the final selections since there are infinite paint choices and easy to get one that complements the other more difficult design elements.

You now have very neutral cabinets so without the brown/beige backsplash your choices are pretty much limited only to what you find aesthetic and what your budget is.

You could go with a bolder more colorful backsplash unless you are afraid of that. My backsplash is a combination of hand fired blue tiles with a lot of movement and some copper tiles to complement my copper sink and ceiling 😀 But my granite is very "quiet" as it is Black Galaxy so not much movement and pattern but the flecks complement the backsplash and the copper accents.

It really won't feel like you've updated until you change both. Just an aside; the two pieces of art flanking the window are too large for the space they're in, and the sculptural piece is hung too high. The panels look a bit skinny and pinched into the space. Remove the art so you can tweak the curtains. A good "rule" for hanging art is 5-5-75. That's centered 5' above the floor, or hung 5" above furniture and no wider than 75% of the width of the furniture beneath it. If you want art on those walls, remember that 25% of their width should be exposed wall on either side of them. Once the curtains are fixed, you can rethink art, if you even need it there.

A plain quartz countertop in one of your backsplash colors