Suppose someone uses a website against its Terms of Service, EULA, etc., such as by using multiple accounts when the website limits users to one account or trial period each (to limit usage), using a website to make profit even though the website says not to, etc., in such a way that there are monetary damages to the site owner. Surely, the website could file a lawsuit to recover any damages and profits made inappropriately, but would the person be committing a crime, such as theft or fraud, or only if the person is providing inaccurate information?
Yes, it might be a crime
The relevant legislation is the Computer Fraud and Abuse Act which, among other things, makes it a crime for a person to "intentionally accesses a computer without authorization or exceeds authorized access, and thereby obtains— (C) information from any protected computer;"
A "protected computer" is one "for the use of a financial institution or the United States Government" or more relevantly "which is used in or affecting interstate or foreign commerce or communication" - the computer you are accessing is connected to the Internet and therefore it "is used in or affecting interstate or foreign commerce or communication" because nothing that happens on the internet is wholly contained within one state.
The question of if merely violating Terms of Service amounts to "intentionally" in the criminal standard was addressed in United States v Drew. That case was about the defendant cyberbullying someone on MySpace, sadly to her death. This was a violation of the ToS but the appeals court vacated a guilty verdict on the basis that breaching the ToS, in this case, did not amount to criminal intent. However, they did leave the prospect open that it might in other cases.