It is a local maximum, just as you suspect.

The definition you're using that claims that it isn't a local maximum might have been misworded, or worded correctly but subtly in a way that you've overlooked. The important thing is that you've obviously understood what's actually meant by a local maximum.

(That you've detected this discrepancy means that you're engaged - thinking, questioning, and reflecting rather than just passively absorbing the material. I'd wager that you're a very good student.)

At the boundary of a domain, you consider the open interval intersect the domain. (Technically, that's what you're doing everywhere, except that distinction between open interval and open interval intersect the domain doesn't have any tangible consequences at most points in the domain.)

Local maximum (minimum) means that, when resticted to domain values that are some fixed distance from the domain value of the maximum (minimum), that the function's value is greater than or equal to (less than or equal to) all other values of the function in that region of the domain. It means that "nearby, the function is never greater (lesser)."