In a debate, should I judge my opponent's premise based on what they say even if it's not what they mean?

You have two different questions here.

In a debate, should I judge my opponent's premise based on what they say even if it's not what they mean?

If you are ever not confident that you know the intent, you should ask for clarification. The concept of "steelmanning" is a good practice; to present the strongest version of your opponent's (interlocutor's) argument. This is similar.

Would my opponent's premise be false even if they meant to use the correct terminology? 

Yes. False is false.

My opinion is that it matters what the circumstances of the argument are, and who the opponent is.

The simplest and most friend-retaining approach is to suggest what you think he meant, mildly,

Yes, the premise is false, because truth is neutral to the intent of the person making the claim. Even if they meant tigers, it would remain untrue until they corrected themselves.

1 + 1 = 5 does not become not-false because I meant to change that 1 to a 4. As long as it remains uncorrected, it is wrong. In an exam you'd be marked based on what you wrote - the erroneous 1 - rather than your intent - the correct, but imaginary 4.