From another thread just down the board: (Thanks @euro cruiser)
Italiarail is a reseller, not a train company.
The two national train lines in Italy are Trenitalia and Italotreno. Only Trenitalia serves Civitavecchia.
IT - Trenitalia
You may not be seeing all the trains because you are choosing a date too far in the future. Most of the trains to Civitavecchia are regionals and they don't load them into the system more than a few months in advance. To see what the schedule will look like when you are there, pick the same day of the week for the upcoming week. Small changes may happen, but you'll get the general idea.
You should use only those two websites. The prices will be best about 4 months out when they first load the fares. Anybody else is a reseller, and while easier to use, will cost more.