That's very perceptive . . . good for you!
You may not need convincing, but I agree with you . . .
For reference, my coordinate system is set up like this: We have a circle of radius 2 centered at (1,2,5) in a plane parallel to the yz-plane.
Are both of these answers correct?
. . . . . . Yes!
I cannot think of a reason why both would not be correct.
The only difference I see is what point represents.
I ask because the back of my book only gives the first.
+ - - - - - y
The positive -axis is upward, the positive -axis is to the right,
. . and the positive -axis "comes out of the screen."
The plane of the circle is parallel to the "back wall,"
. . so the circle is facing us.
With your first equation, starts the circle at 3:00
. . and generates the circle counter-clockwise.
With your second equation, starts the circle at 12:00
. . and generates the circle clockwise.
Either way, we get the same circle.