What is the difference between utility, payoff and expected utility, or are the terms interchangeable?

I've started teaching myself game theory recently, but so far I haven't come across anything clarifying these terms . This is my understanding of the terms based on what I know:

• Payoff = Utility. Both are interchangeable, and represent a cardinal preference ordering for a given player. Hence payoff/utility must be a natural number.
• Expected utility/payoff: The expected utility is the utility for any given pure or mixed strategy. Hence expected utility could be a real number.

But I am still unsure, because this definition given in the textbook I'm using seems to use utility function and payoff function as though they are not interchangeable:

The only difference seems to be that the utility function takes an outcome as an input, and the payoff function takes a strategy profile. I've seen 'utility' and 'payoff' used interchangeably as terms, but perhaps they both describe different functions? I am also not sure how to fit this information into my understanding of expected payoff/utility.

Please could somebody check my understanding and correct any misconceptions?

"Utility" is a foundational concept, a nomenclature label to name what we experience from our interaction with tangible things or intangible phenomena, situations etc. So e.g. "consumption of a good generates utility for me", "hearing about a natural catastrophe creates negative utility for me", etc.

A Utility function is a mathematical toοl with specific properties that assigns a real numerical value to the above psychological/subjective states. In its simplified form, its argument is monetary wealth, say $$U = U(w)$$.

"Payoff" is a loose term, used for "what I get out of something". In some cases it is "quantified utility", but it can be directly monetary. It can also signal the general situation ("how are payoffs structured in this game?")

"Expected Utility", is, to begin with, a general term to point towards the Expected Utility Theory, which makes specific assumptions about how in an environment of uncertainty a rational person quantifies their utility.

In its narrow sense, it is the expected value (1st raw moment) of the utility function, $$\mathbb E[U(w)]$$ where here the argument of the utility function $$w$$ is a random variable, so the utility function is also a random variable, and its expected value, is a real number.

Players in game theory are assumed to have preferences over the possible outcomes of a game. Players are also assumed to adhere to the axioms of Expected Utitilty Theory, so in the presence of risk, their preferences over lotteries over outcomes can be represented by a (von Neumann-Morgenstern) utility function which is calculated as an expected (Bernoulli) utility. Given a profile of (mixed) strategies, a player's utility of the lottery over outcomes generated by this strategy profile is called his payoff. (Some texts say expected payoff to stress that it results from taking expectations over payoffs from pure strategy profiles, but this is not necessary.)

So yes, payoff functions and utility functions differ just by what they take as input and are therefore more or less interchangeable if you tolerate some sloppiness. The only thing certainly wrong is your assumption that payoffs or utilities have to be natural numbers. They can be any real numbers.