Hi anuta,
There is no problem at all, except that you must bring your marriage certificate with you.
Book your tickets using the name in your passport.
You will need to show your green card to the airline agents in Russia, because they need to know you are allowed to enter the US before they can let you on the flight.
As your green card is not in the same name as your passport, your marriage certificate is required to prove that the passport holder and green card holder are the same person.
Don't bother changing the name in your Russian passport, but if the Russian embassy in the US can annotate your passport with your married name, that can only help.
Your "source" is a fake passport application site.
From the horse's mouth:
Thank you so much! To change a name in my Russian Passport is such a problem - I would need to fly to DC and then wait, the presence is needed and don't have enough time to travel there.