Unable to send e-mail

Hello community,

Luckily, there are only a few topics dealing with mail problems... I've read them all, but could not find a solution:

My backdrop cms refuses to send e-mails... user registration, password reset, webforms: "Unable to send e-mail".

The log messages do not reveal much: "Error sending email (from ... to ...)"
The server logs do not have entries about mail attempts.
Te spam folder in my mail client are empty.

This is what I tried so far on two different installs (1.27.1 and 1.28):

1. Using php mail function, at admin/config/system/site-information I tried two different mails (info@real-domain.com & info@other-domain) – same result:

When reseting password: "Unable to sent e-Mail"
When creating an new account: "Unable to sent e-Mail" AND "A welcome message with further instructions has been emailed to the new user". But no mail is coming in.

2. Using smtp authentication support
More or less the same behaviour:

• Blue message: 
SMTP ACTIVE: Mail delivery is handled via SMTP module.
SMTP ACTIVE: Mail delivery is handled via SMTP module. (yes, twice)

• Green message
The configuration options have been saved.
A test e-mail has been sent to test@test.com. You may want to check the logs for any error messages.

• Red message
Unable to send email.

The recent log screen shows "Error sending email (from ... to ...)"

Does anybody of you has an idea what else I could try and test?

Thanks in advance and best regards