Tyrrells Premier games are appointed by the RFU. This could see anything from a L3-L6 referee and occasionally a L7 if somebody is being fast-tracked.
Tyrrells Premier Development (2nd) teams are appointed by the local society so could be officiated by anyone that the appointment secretary feels is up to doing the match, regardless of actual level to some extent. (At least in our society.) In general it is mostly officiated by L7-9 referees. Women’s games outside of this get a similar level of referee depending which league they are in and who is available.
Linking an actual level to these games is difficult since the womens’ game has been developing at quite a rate over the last few years. Over the years the level of referee appointed hass been raised because the coaches and clubs want more technical accuracy in the refereeing so that they can coach to that level and help development.
Welcome to the forum.
I wouldn't worry too much about the report from the loser.
Tyrrells Premier games are, as outlined above, invariably refereed by RFU appointees usually NP @ Level 3-4. My lad is required to do them and he is L4 - they also have ARs appointed too. I did a pre-season game for two women's teams, one of which turned up with 12 and borrowed a few from the home team. It was pegged as L6 or L7.
In reality no L6 or L7 men's team would turn up shorthanded but the development of the women's game needs (for reason's outlined by Balones) referees at a Level to provide accuracy and management that facilitates development and not absolutely nailed on to the level relative to the equivalent men's game. I'll say no more.
As you referee mens' and womens' Rugby it will be come apparent that they are different and present different challenges. As you progress, as I'm sure you will based on the thread so far, you'll know what's required.
We have a couple of women refs in Yorkshire who are progressing well after starting quite recently (relative to some on here) and the opportunity to progress for women referees is there if you're dedicated/good enough.
Enjoy your refereeing and ask as many questions as you need to - you may get a concise, accurate answer - occasionally.
Cheers