My understanding is that the lawyer was authorized to act on the behalf of one party, and the other party was happy to sign the agreement that they were sent.

A signature is merely evidence. It's neither sufficient or necessary for agreements. It just happens to be very strong evidence so we tend to think of it as the absolute sign of agreement but legally that isn't the case.

Clicking "send" or "buy" is signing stuff nowadays.

Well, in most civilised countries; perhaps not in the most backwards of them.