I guess to my brain a definition based on wage labor just seems more straightforward and more clear cut, ie what WoofWoof91 said below. Here's what I mainly found unclear:

If you got your paychecks based on other people's work,

This feels wishy-washy, what counts as "based on other people's work"? If you structured a company where the managers compensation was based on the output of the workers they supervised, would that make the manager a capitalist? Maybe I'm just misreading.

who you have a stake in determining the way production is done

This I actually don't understand.

and from your reply, I think this line is clarifying as well:

They still need to perform their own duties and are held accountable by their own bosses.

or to take a less descriptive approach, it seems simpler to say, capitalists are the ones who own the enterprise. Since that ownership is what gives them impunity/autonomy to do with the enterprise as they see fit. And I like WoofWoof's comment because it also addresses the degrees of ownership (people who derive some income from owning capital, but a minority of their income)

Sorry if this is annoying lol