Whenever I want to search for a string I find it 50/50 whether I want to search for the string under the cursor or not. So instead of requiring the cognitive overhead of having two different keybindings for live_grep and grep_string, I'd suggest combining them (as a third option).

So when you search for a string it takes the string under the cursor, but has it grayed out like an auto complete suggestion. You can then easily press enter if you want to search for that word, or simply ignore it and type whatever.