Wouldn’t 82% of negative cases being correctly identified mean that 18% of negative cases were incorrectly identified as positive(false positive), or in other words it will mistake 18% of healthy people as having Alzheimer’s, but will always identify an Alzheimer’s patient?

82% true negative rate == 18% false positive rate. Your description is correct. It's likely that those statistics will be somewhat worse when the algorithm is tested on a broader and more diverse test set, which the authors claim to plan to do in the future.

Thanks for digging up the paper, false positives are probably more acceptable than false negatives in this scenario (wrong positive diagnoses can be rectified with additional tests but a negative diagnosis will often not be confirmed by additional testing).

Both things matter, and you need to consider what the follow-up test might be and what it would cost not just financially but also in terms of pain, lost time, risk of harm and so on.

There are medical tests which have such a high rate of false positives that in practice in the absence of other indications you would ignore a positive result, because the only possible follow-up test would be a non-risk-free biopsy, and therefore, for your peace of mind, it's probably better not to do the first test at all.

I think OP initially mixed up the numbers but now it's correct (18 % false positives).

A false positive rate (currently) that high may preclude this being used as a widely available diagnostic test, but if used within a patient base already showing other signs of problems it might be more acceptable. After all, if you're going to see a memory specialist you're already thinking there's a problem.