Too late. I took the visa center's airport transfer partner (
http://taxivisa.com /
http://transprestige.ru ) and it was really not worth it.
I chose the "comfort" option because their "business" option was overpriced, and the comfort was basically that of a regular taxi. And then they didn't accept credit cards, even though that was clearly part of the deal. Bad experience.
For the same price we could easily have ridden Uber Black - which we did all the other time, also back to the airport.
My suggestion is therefore clearly to ride Uber Black in Moscow, it's not expensive, like a regular taxi in most other cities.
We didn't bother with the train. Traffic was perfectly fine at the time we rode.