How about, "elapsed time"?

I'm not sure about elapsed time. It's not necessarily the case that any time has elapsed--it's merely an offset that can be set to any value. Perhaps just "time offset"?

Elapsed time of zero is a valid elapsed time. The text in this paragraph is somehow unclear, I would assume that current_time = now - timeline.start_time, that for me is exactly the elapsed time, as its always >= 0. Negative can't be, in a well designed system.

It's reasonable for the "origin time" to be negative so perhaps "time offset" fits that better.

It could be, but it sounds general, "origin offset time"?

Maybe "relative time"?

Oh, I like origin offset. Thank you!

In my experience from TTML, it is massively helpful to have terminology distinguish between spatial and temporal coordinate systems, because it is extremely easy to get confused between the two. It is typical in English language usage for temporal concepts to be described in spatial terms.

For that reason, I would advocate for example "origin offset time" to "origin offset".

However I would also note that in temporal coordinates, an offset is typically a duration whereas a time is typically a specific moment.