In particular, I am teaching a course this spring based on two Springer books that I was able to download yesterday. If I e-mail PDFs to all of my students, am I legally in the clear?
I am quite annoyed by this revert -- especially by the fact that big publishers can generate so much good will just by a simple error that gets shared on the Internet, while the correction will get minimal exposure.
1.2 You may solely for private, educational, personal, scientific, or research purposes access, browse, view, display, search, download and print the Content.
1.3 You may not: [...] re-distribute, reproduce, or transmit the Content by any means including electronic (e. g., via e-mail) nor post it on their personal or public websites or on public networks...
So while you may have the copy of the book legally, it seems unlikely that you can give it to your students. There's always a chance that what they're saying isn't legally enforceable (see non compete agreements), but I don't think that's the case here. I'm not a lawyer, so I can't say for sure.
If there's no license specified, default is "all rights reserved" and you cannot distribute the books to your students free of liability.