@Harambe
Pretty certain botanically speaking your first definitions are correct. Other criteria or terminology can develop to decide if something is a fruit/vegetable in a colloquial or nonbotanical sense, but that's generally where things get inconsistent.
I.e. people often refer to gourds and squash as vegetables due to the planting methods (among other factors). But melons, which would also qualify as vegetables by that reasoning, are still colloquially considered fruit due to their look and flavor. And ironically it wraps back around to the botanically correct usage.
Usually people who get pedantic about fruit vs vegetable are almost always referring to the more well defined botanical sense, (strawberry = vegetable, melons and gourds = fruit), so it's weird that he would apply that reasoning only for the strawberry? Though maybe that's the joke.
Regardless, I would add a caveat to "use whatever you want": use either so long as you're aware of the context you're using it in. Notably, if you say two things are in the same category, by using two different definitions of said category, you're just being unnecessarily confusing and/or obtuse (as I assume was part of the joke - unless the author legitimately didn't understand the distinction).