The Authentication field has to be set to "disabled" in our case. No SSL, no login or password needed.
If Authentication is set to AUTO (which I thought would include the deactivation if realising it's not needed) it won't work for servers which don't have an authentication.
Ok, now we're all happy here and next time I'll try the easiest ways first. The problem was that the Toshiba technician told me I'll need mail server data, different from the usual email-account's data.
So I tried it with data that our webpage deals with to connect to the same smtp server, but also the simple things. I think the authentication was set to AUTO by default. Bad for us!
thx to everyone advising me here