On the demo site I took the following route and saw no issues

• change the Contrib page to accept Pay Later
• completed a Pay Later membership for the demo user
• confirmed that by then clicking on Edit for membership at backed, clicking on the Contribution block with 'more' and 'record payment' the membership was set to Current and there was no 'renewal' Activity on their record.

Hope that helps

Data will be on https://dmaster.demo.civicrm.org/civicrm/contact/view?reset=1&cid=203 for less than 12 hours till it is flushed.

On the demo site I took the following route and saw no issues

• change the Contrib page to accept Pay Later
• completed a Pay Later membership for the demo user
• confirmed that by then clicking on Edit for membership at backend, clicking on the Contribution block with 'more' and 'record payment' the membership was set to Current and there was no 'renewal' Activity on their record.

Hope that helps

Data will be on https://dmaster.demo.civicrm.org/civicrm/contact/view?reset=1&cid=203 for less than 12 hours till it is flushed.