As I understand, the definition is as follows:
"The tyre x-axis is an the intersection of the tyre
centre plane with the road plane, assuming a flat road.
A tyre slip angle is an angle between the tyre x-axis
and a direction of wheel travel."
The Internet definition is vague.
You can not measure "the direction the tyre contact patch
with the road is pointing", really.
Even when the tyre is free rotating with no slip angle,
most part of the tread in the tyre contact patch is moving
laterally. (The tyre contact patch is not rectangular.)
So only a small part of the contact patch is travelling
in the same direction as the wheel.
The DVD definition may have mixed a tyre slip angle
and a vehicle slip angle
It is correct when the yawing velocity is zero so that
all the tyres are travelling at the vehicle velocity.
(e.g. big corner R, or a massive final understeer when
the car goes dead straight, no yaw velocity...