Caulk

Caulk.

first tape off the 'seam' with some blue painters tape so you get a clean finish.

top and bottom... yes, it takes a minute, but you'll get a nice clean, narrow finish

Remove the grout in the joint first. Then caulk.

Do it fast, too. This happened to me and was one reason we had to do a renovation. What a mess water does.

It’s best to remove the cracked grout and replace it with caulk. Caulk is more flexible and better suited for areas where the countertop and backsplash meet, as it can handle movement and prevent future cracking.

Silicane caulk is the right choice but now you very carefully need to remove that grout and it will be a PITA to do but worth it. I would not do caulk over that mess it is just looking for a place for mold to happen under the caulk. While you are doing this why not get a new less dated backsplash too