How to change the rates?

Make an adjustment on the booking before you confirm. It would be better that way. Otherwise, changing your rate table means the lower rate would be available to other customers.

FYI, customers who haggle about price often turn out to be major PITAs. You already have a very low rate for house sitting. Unless you are just sleeping there, providing 24-hour care for that price is already a huge bargain. I realize you are attempting to book your first client, but you should put a value on your time.