In other words: is there a policy that a document used in a train booking must also be acceptable for entry and exit from Russia?
No, there is no written policy for that. The only thing you should be aware of is the conductor during the board check validates your document and ticket. The document number and your names in it must be the same as in your ticket. So, you must provide them your ID-card, and, as far as I know, it doesn't looks like a passport. This can be a problem in general, as the conductors simply can be unfamiliar with such document formats. This can cause the delay in boarding and even the denial for the trip (as you said that you do not speak Russian, they will try to find an English speaker).
From other side, the Sapsan's are frequently used by foreigners, and for this exactly train you really not in a danger. But I still do advice for you to get the passport with you, for the safety (it much more safer to bring it with you rather than leave it in a hotel).
Passport check are rare in general, but on railway stations and subway they still can be met. In case if you have no passport with you, police can take you to the police station for a further checks, and they can leave you there for hours. This is not the situation I want to be in.
As for the difference between domestic and international trains - I think that you really should use the passport for international trains, as the conductors during the boarding are checking the visas for all the passengers, and there is a huge probability of denied boarding for such trip if you have used not a passport (even if you have it with you) - they simply will want over-insure themselves.
Also, be aware of that the Russian Railways site may not accept your Visa/MasterCard payment card if they were not issued in Russia. So, you have to find another site for ticket purchasing. For example, this can be used:
but personally I haven't used any of such sites for myself.
Kazakhstan is a member of CIS, so you can use your ID as it is
ЗЗ - Foreign document (passport, identity card, etc.) issued to citizens of the CIS
But I suggest you to use it only for domestic trains, as I stated before.