Unless it’s a careless mistake, these two cells appear to me to be equivalent. The only difference is, in the first cell, an argument is explicit as opposed to a previously defined name in the second cell. Using Evaluate[] and/or ReleaseHold[] on every possible argument in the explicit example doesn’t change the cell’s output. I think the second result is the correct one. Using the default lexicographic monomial order, I think the remainder of f divided by the Groebner basis is 2z. As of 18 Oct. 2021, 119 people had viewed this with no reply, which is unusual in my limited experience.
This question has been answered with generous help from Wolfram tech support. Contrary to my initial statement of the question, Mathematica produces the correct result if Evaluate[] is used correctly. Specifically Evaluate[] must enclose the whole argument as in the last cell in the attached example. I’m still not sure why Mathematica returns True when Evaluate is used incorrectly.