I've seen different sources saying that "one needs to register in Russia within the first 7 working days", and other sources saying that "one needs to register in Russian when staying for 7 working days or more". Those may sound like just the semantics, but they have two different meanings. What is the actual rule, and do I need to worry about registration in my case?
Per WayToRussia.net, which isn't official but is usually pretty darn spot-on, and cites the law in question as a reference:
Every foreigner who comes to Russia should have his/her Russian visa registered within 7 working days upon arrival, excluding holidays and weekends.
So you're theoretically OK. But they go on to say:
Before 25 March 2011 you had to be registered within 72 hours upon arrival (excluding weekends and holidays), so it might be that not all the authorities are aware about this new rule yet ... we strongly recommend you to register your visa in 72 hours (only working days count) upon arrival.
So it's probably worth doing anyway, just for peace of mind and/or in case your schedule changes (what happens if your flight is cancelled?). Since registration is actually the responsibility of whoever you're staying with, this should not be much of a hassle.
The rule is that you have to be registered within 7 working days of your arrival. Source from the Russian Migration Service, referring to foreign citizens temporarily residing in Russia:
временно проживающий или временно пребывающий в Российской Федерации иностранный гражданин - по истечении семи рабочих дней со дня прибытия в место пребывания
a foreign citizen temporarily residing or temporarily staying in the Russian Federation - after seven working days since arrival to the location of their stay
However, the registration is not your obligation if you're staying as a hotel. The same page goes on to list exceptions and says
находится в гостинице или в иной организации, оказывающей гостиничные услуги [..]
is located in a hotel or another organization offering hotel services [..]
If you are going to be staying at a hotel or similar, then the hotel will handle your registration. You will already have the necessary paperwork, as they need a passport and a migration card, which you will receive just before you enter the country.
In your case, if you're leaving before the 7th workday, you're fine without a registration, but your hotel will probably register you anyway.