I'm not a math teacher, so there may be a technicality I'm missing, but you're not wrong, they essentially are the same. 2 rays can have the same endpoint, (like hands of a clock, say when the clock has one hand on the 9, the other on the 7) but they are only considered "opposite rays" when they wind up forming a single line, each ray of the pair pointing in opposite directions (like when the hands line up to show one hand on the 7 and one hand on the 1).
One way of thinking of a ray is as a "half line" Opposite rays are the two halves of the same line put back together, so in the video you watched, the opposite rays YX and YZ together could form the line XZ. To be proper in a math class, you'll have to imagine I put the single headed arrow above YX and YZ to show that they are rays, and the double headed arrow over XZ.