What people actually use the term incorrectly for is the internal tracking necessary to manage the process of maintaining a household and relationship, including doing chores.

The first version is a strawman because no one thinks that the straightforward action of chore-doing is also, secretly, emotional labor.

The stronger version makes it more clear why any person acting in good faith would try to apply the term at all: the managerial work is unseen, mostly unacknowledged, internal, and as a bonus it's emotionally taxing under many conditions. That is to say, it has a lot in common with the real concept of emotional labor.

So it's true that the wrong term in being used, but a stronger understanding of what the incorrect people are trying to refer to makes it clear why there is a problem here to discuss in the first place.