Update.
I finally lost all Internet access again and - long story short - an independent techie found my modem had been reconfigured (no-one here even knew how to access the modem). Instead of the provider's login and password, my wife's e-mail address and some asterisks. He restored the appropriate entries and access was restored.
Next morning, lost access again. This time I accessed the modem and, scanning through the entries (the changes made previously stood), was intrigued (as DNS server had been one of the problems identified by Windows network diagnostics) by "Manual DNS Server" and the two IPs entered. The technician was too, and suggested replacing them with Google (8.8.8.8 and 8.8.4.4), which restored access. All hunky-dory to date - including apparently clean bank access.
I then googled with the two IP addresses, which turn out to be hosted in the Russian Republic.
That is, my modem had been reconfigured somehow to two Russian DNS Server IPs.
Go figure.