Yeah, taking the lead I think is a pretty common expression, meaning that you’ll take the initiative, but I’ve used " taking their lead" to mean that another person has taken the lead and someone else is following them.
Which is apparently not real at all, but I only became aware of this because another Lemmy put up a TIL post that explained how they thought that was an expression and discovered after using it their entire life that it was not in any dictionary.