Is economic profit just excess profit?

Your confusion has to do with two notions of profit -- one from accounting, one from economics.

In accounting, one of the statements issued by businesses contains information about their income and their expenditures. Without going too far into details, they compute both of them on the basis of commitment -- they register the transaction when the deal is concluded, not when money enters or exists their bank account -- and you see the details of expenditures, reducing income. What is left down, after tax, is what most people think about when they talk about profit -- they really do mean to say the difference between what has been earned and what has been spent.

In economics, given that the intention is not to keep track of transaction, but to study the choices of people, the definition is slightly different. It's whatever amount that is received in exchange of the bare minimum required to maintain the productive capital in place. So, in the accounting profit, you have two things: part of it is the price of capital and part of it is actual economic profit (of course, you'd have to re-do some of the maths accountants did because some of the expenditures they write down are not suited for an economic analysis). Economic profit could be null -- or even negative --, while the accounting profit is positive.

Another detail worth noting here is that owning a business does not earn you nothing when the economic profit is null -- you earn the "wage" of your investment in productive capital. And, if you provide labor to the business by administrating it, it's part of the expenditures -- and accountants did calculate it properly -- and you've earn that as well as your return on investment.

We can't say profit is 'excess' unless we define *relative to what*?

There is no such thing as normal profit, because profit is always a response to non-normal circumstances. In the imaginary construct of an evenly-rotating economy, in the imaginary construct of economists' "equilibrium", there is no profit, and no loss.

Why not? Because in that imaginary stasis, people keep repeating the same purchases over and over again. Therefore investors take their capital out of activities that are making losses because the consumers don't want them, and invest in activities that are making profits because the consumers want more. This tends to eliminate profits.

In reality, this doesn't happen, because new data are always entering the market: rain falls, people are born, new inventions are made, crops fail, roads degrade, new ones are built, and so on.

Profit results from the activities of entrepreneurs adjusting the factors of production to these new gaps between
a) where the factors of production are, and
b) where the mass of society want them to be in order to satisfy their most urgent wants.

Therefore we cannot say there is such a thing as "normal" profit because there is no normal or ideal state to compare it to. Profit arises from adjusting the factors of production to gaps between the real and the ideal state.

Economic profit is a profit = total revenue - opportunity cost. The opportunity cost includes implicit and explicit cost or accounting cost. Normal profit is a situation when economic profit is zero.