Is this general relativity diagram correct?

The diagram is correct. However, you may be misinterpreting it in terms of the equivalence principle.

Also, why are the two reference frames presented there equivalent?

You are correct. These two reference frames are NOT equivalent. The easiest way to see that they are not equivalent is to note that an accelerometer at rest in S will read 0 but an accelerometer at rest in S’ will read g in the upward direction.

However, although they are not equivalent they must be consistent with each other. In other words, if a detector near the ceiling detects the laser in S then that same detector must also detect the laser in S’

I can't really see why the photon would travel following a straight line in the first one

So let’s think about a reference frame that is equivalent to S. In this case an accelerometer at rest in S reads 0 and there is gravity. So the equivalent reference frame would be one where an accelerometer also reads 0 and there is no gravity. In other words, an elevator adrift in interstellar space far from any gravity. Such a frame is clearly inertial and so light will travel in a straight line in that drifting elevator.

Since the drifting elevator is equivalent to S and since the light travels in a straight line in the drifting elevator it also travels in a straight line in S. And since it hits the detector in S it must also hit the detector in S’. Therefore light must curve in a gravitational field.