I think it is clear that MathSciNet's terms of use were written by the AMS's lawyers, and not by anyone with any sense of how the database is used in real life. By the letter of the rules, they would also appear to forbid using MathSciNet results to produce citations for a paper which is published or otherwise shared; but this is probably the most useful feature of the database. (Indeed, why else would they have an option to generate BibTeX?)
It would seem to me (though I am not a lawyer) that quoting a brief passage from a review would certainly be permitted by "fair use" exceptions in copyright law (in the U.S., for instance). AMS shouldn't be able to go after you on copyright grounds. They might be able to go after you for violating the terms of the contract, but it's hard to see how they'd argue any damages, though they could still revoke your subscription to the database. However, I can't imagine them actually doing such a thing, and if they did, it would undoubtedly provoke a major outcry from the academic community (which after all makes up AMS's membership).
In short, the terms of use as they currently stand are absurd and should be changed. As an AMS member, I think I shall write to the leadership and tell them so. I'd encourage other AMS members to do the same.