Before attempting to start any PuTTY sessions, the initial WinSCP connect and login was failing because it had the wrong host name. The host name was edited and replaced with the correct current host name. This allowed the login to proceed correctly, after which WinSCP operations worked, including "Open terminal".
However, even then "Open session in PuTTY" would fail to connect. This seemed to be because it persisted in attempting to connect to the wrong destination host name. Though I cannot establish this, I suspect the host name it was using might have been the earlier obsolete host name that was failing for WiinSCP before it was edited and replaced with the current host name. In other words, that change doesn't seem to have propagated to being used by "Open session in PuTTY".
I used "Clone to a New Site" to duplicate all settings, but I did not make any changes. Supposedly that should have been identical. However, the operation of cloning the settings appeared to create a consistent set of specifications for the host name. In the new cloned session, "Open session in PuTTY" worked as expected.
Yet even after cloning a new session, the original session from which it was cloned still failed to be able to "Open session in PuTTY" despite having exactly the same WinSCP destination and the same login specifications. It appeared to still be insisting on going to the wrong destination host name.