In video 5, why is it important to define "dominant strategy" as "this line always gives the greatest value" as opposed to "greater or equal value"?

Would it be somehow wrong for Player One here to say "I don't care what happens, I pick B, because there is never a reason not to"? If not, then why treat this case differently?

EDIT: Okay, I got it. There is no difference for Player One, but may be a difference for Player Two in that they might be unable in some situations to predict Player One's move (which doesn't influence the Player One's payoff in such situation, but may influence Player Two's payoff).

If this is also your reason, it might be useful to mention the bit "...and Player Two can predict that a rational Player One will choose their dominant strategy" in the video.

EDIT2: Or maybe you should introduce the term "weakly dominant strategy" immediately after explaining that "greater or equal value" is not a "dominant strategy". Just to make it clear that this type of situation will not be ignored later.