I double-checked the "Send email from" value, making sure it was from the title of my website, and not from a specific user or post author. However, I just posted a new post, and got my own name announced in the notification e-mail again.
I then tried sending a message to the only registered user (myself) using the Send Message function, and got the correct "From" address in the mail.
Then I decided to set the "Send email from" value to "post author", and guess what, I got the following error message "Message failed! Check your settings and check with your hosting provider".
As only my master e-mail address was involved in the erroneous messages, and the master address is only my identification with the hosting provider (not used anywhere else and cannot receive mail), I realise it does have to do with my hosting settings.
Just got off the phone with the hosting provider who suggested that Subscribe2 found the system e-mail address somewhere in my WP installation files and is using it for its e-mail function.
Any idea where the plugin might have found it?