A not so technical is just use one survey with branching or display logic and if they branch/display into the section set embedded data for opened and if they branch/display out of the section set embedded data for completed/finished.
A more technical approach could use the update response API. Once the first survey finishes, carry over the response ID to the second section, when they complete it you can use webservice to update embedded data in the initial survey for started and completion.

Thanks for sharing.
I don't think the first option will work in this case because the "do not record" setting overrides any other end of survey setting if that specific end of survey is reached in the survey flow, meaning it doesn't send anything whatsoever to the database. This appears to include embedded data, as well as any data entered prior to this module/block (suggesting that the "do not record" applies to everything, not just what's in that logic branch in survey flow), which we do need to have in this case.
I'll look into the 2nd approach with separate surveys though and report back here.

In the first scenario, you are using one survey so you wouldn't want to enable the "Do not Record" option. In order to not record it the data specifically, you can set the answers for those questions to null at the end of the survey before the end of survey block in survey flow. Just another way I guess.