This is a timeline UI bug and not a REST or data problem. I verified it in 7.9.3 and 7.10.2 as well. It looks to be fixed in the upcoming very soon to be released 7.11.0
The rule.name is in the JSON and shows up as columns but not when you search it:
In the upcoming 7.11.0 where the view of the data is going to be on the right side, it looks like timeline has it fixed. I looked but could not find the PR for this fix though. It might have been fixed along with other bugs:
I don't see any plans for back-ports for timeline ui for this bug for 7.9.x or 7.10.x at this time but you shouldn't have data loss when you upgrade to 7.11.x after it is released.