Dunkred
I think you are confusing two different things that "seven working days" refers to. a) It is the minimum length of stay in Russia before which any registration is required. b) It is also the amount of time that a private host has to register a foreign guest.
These are not referring to the same thing. If you stay somewhere new after day seven of your total stay in Russia, your new host still has seven days to register you, but they must register you for each day eventually (because by that stage you will already have been in Russia for more than seven days). Confusing, I know.
So the 7-day window does not mean that if you stay in St Petersburg for six days, go to stay in Moscow for a few days and are registered by a hotel, and then go back to St Pete for another six days, you never need to be registered in St Petersburg. Once you have stayed in Russia for more than seven working days in total, you then need to be registered at each place of stay (including, therefore, your second stay in St Petersburg).
I also think you are wrong in thinking that a tenant (your partner) can register you instead of the landlord. Try asking AI about this.
When I asked, it said (of 109-FZ, Clause 7, Part 1, Article 2): "The requirement for ownership is primarily derived from the phrase "in accordance with the legislation of the Russian Federation", which links this law to the Russian Civil and Housing Codes. Under these codes, a standard tenant usually lacks the legal authority to grant usage of the property to another person, as that right rests with the property owner."