As far as I can tell, the filter=5 for posts on /user_actions.json?offset=0&username=USERNAME&filter=5 does not correspond to the Posts in the report. It looks like the Posts on the report includes both the posts (filter=5) and topics (filter=4).
I’m assuming that posts are supposed to be replies + topics. It would be nice if someone more knowledgeable could confirm this.
If this is the case, I can try to use replies (filter 6) and topics (fliter 4). However, I’m a bit concerned about this as the report in the Discourse manager is giving the correct, expected numbers.
Try and search for an example of using the Discourse API to produce reports.