I would like to have an importance level which only sends email, and not SMS (or chat message btw.). Currently even the lowest, WARNING level, results in SMS messages. I find SMS quite intrusive. For example I would like to get an email if a disk needs to be extended in the future, and a phone call if it is full or it is expected to be full within hours.

According to the help message on the Importance field:
"Severity level of a failure. Critical alerts are for failures you want to wake you up at 2am, Errors are things you can sleep through but need to fix in the morning, and warnings for less important things".

According to this description warning indeed should not result in an SMS, but it does.

The workaround is to duplicate the service in question and add only email to the Alert types list.