I can chime in some guild info since I've been doing wars actively since day 1.
You are correct, no guild can perform pray during the first phase of the war. I assume it's to prevent a guild from stacking pure defense at the start so they become impossible to attack.
You can pray in any other phase of the war, as well, once an hour has passed since a previous phase has ended (except after the final phase) a guild member can pray up to 5 times between phases, probably so those who cannot make the timeframe for the actual phase can still do something to help their guild.