Further, your definition doesn't include situations in which there is no employee (such as the one I gave).
It is more correct to say that the profit is the gains made to some agent in a market transaction.
That's why I asserted that profits are nothing more than unpaid wages. Since it's objectively a trade of your life for wages, paying others less than the value you produced is simply not valuing human life. The worst sort of theft.
For your example, it doesn't hold up because most people only make enough to live, there's nothing to optimize other than going "full ramen". While others have far more than necessary for basic food, shelter and medical care. Those who are just-surviving aren't doing anything wrong, when an economic ponzi scheme simply doesn't value their life by paying them less than the value they produce.
And yet, both the planet and individual people are far wealthier today than when they were producing and receiving "full value" for their labour.
This is ridiculous. As someone who has, in the past, run their own business, and now works for a living, the fact of the matter is that my employer does a lot of work I am unwilling / uninterested in doing, such as making sure there is money to pay me and advertising the product to others, as well as figuring out how to accept payments, get the requisite insurance etc.
The profit retained by the company is the share agreed upon in advance paid to those who, through their own labor, at some point made the company possible, whether it be by direct action, or by stored labor output from a previous position (i.e. capital).