should I change that name in the parameters as well ?

class rsyslog (
<...>
  $daemon              = 'rsyslog',
  $daemon_ensure = 'running',
<...>
) {

would get:

class rsyslog (
<...>
  $service_name              = 'rsyslog',
  $service_name_ensure = 'running',
<...>
) {