I've been subscribing to tsunami warning system emails since the mid or late 2000s. They send the first email about the earthquake as a warning that something happened. Then after ,if a tsunami isn't detected they send an email saying that. If there is a tsunami they will send the first warning and as soon as sensors and satellites start to track the wave they will update at intervals with a table of expected arrival times and magnitude or height. So, yes, they send a warning that something happened, then they send information if there is a threat.

Here is an example of the first message sent 9 minutes after 2011 Tōhoku earthquake https://imgur.com/a/1mwAKqc.