I inserted "agreement" in parentheses to clarify the meaning of contract as a type of legally binding agreement here as opposed to other usages of the word.

I don't think this is a controversial definition of the word contract (for example Wikipedia's entry for "contract" states, "A contract is a legally enforceable agreement...").

I did not say that contract and agreement are synonyms that can be replaced with each other in all circumstances, so I'm not really sure that your comment is relevant to what I wrote.

I was also trying to clarify that the employment contract might be titled "Employment Agreement" or something like that but is still a "contract."

But employment minus consideration is either:

If the worker isn't receiving consideration: Slavery, which is illegal in the civilised world (Americans still enslave convicted criminals...)

If the employer isn't receiving consideration: Some sort of payroll error? Are you mistakenly still paying people you fired? Call HR now.

Sure, but the larger point, that employees have employment contracts in at-will states, is still true.