CTRL + W usually deletes everything until the previous whitespace, so it would delete the whole '/var/log/nginx/' string in OP's example. Alt + backspace usually deletes until it encounters a non-alphanumeric character. Be careful working CTRL + W into muscle memory though, I've lost count of how many browser tabs I've closed by accident...
You're telling me!!!
(I use vim daily, with multiple splits in a single instance.)