The value that you've got is number of searches required in Worst case by BS. Each function call is responsible for only 1 search. And your RR justifies that. So the RR you wrote is for number of searches. Indeed we require logn +1 searches if its a worst case. Take n=7,8,any and try. :)

The value that you've got is the number of searches required in the worst case by binary search. Each function call is responsible for only 1 search. And your recurrence relation justifies that. So the recurrence relation you wrote is for the number of searches. Indeed we require $\log n +1$ searches if it's a worst case. Take any $n$, say $n=7,8$, and try and see for yourself.