Me contacté con una de las agencias que mencionan aquí en el forum que preparan la invitación (http://www.visatorussia.com) y esta fue la respuesta por parte de ellos:
"Tourist visa support (a.k.a. letter of invitation) consists of two
documents: a confirmation and a voucher. The documents will be sent via
e-mail. The original is not required.
Please note that Volgograd region is under strict passport control. If you plan to stay at a hotel, we will be glad to provide you with a tourist
letter of invitation.
If you plan to be privately accommodated, please be advised that you may meet serious problem upon arrival. You will need to register your visa and it may be difficult for you to register your tourist visa at a private place.
Unfortunately the local migration authorities do not believe in tourists
staying at private apartments.
Though the registration rules have recently changed, the authorities may
still proclaim that your visa type does not match the actual purpose of
your visit. This could lead to your being fined or placed in one of the black lists maintained by the Ministry of Foreign Affairs, which might make future travel to Russia difficult. In addition, the receiving entity (that is, your actual host) will also be fined.
If you do not plan to stay at a hotel, we definitely recommend you to obtain a private invitation from your friends and apply for a homestay visa.
This takes long indeed but you will be quite safe upon arrival."

Ahora, para quien quiere hacer un primer viaje a Rusia, y que te respondan así, ya no se si te da muchas ganas de "arriesgar" a sacar una visa de turismo y después haber problemas.
Alguna sugerencia?
Muchas gracias a todos los rusófilos de este forum por las respuestas/ayudas que he tenido anteriormente.