Hi Sophie,
Sorry it took me a second to reply to this, I wanted to be certain I was giving accurate information so I jumped into the source code to check how it is implemented.
Checking src/program/time_series.cpp on the develop branch the time_series function is broadly implemented as follows:
1. Initialise the simulation via a checkpoint file or an input equilibration temperature
2. Output data
3. Loop through the simulation for the equilibration time, calling the vout::data output function on each step which internally handles things like output rate
4. Reset temperature and statistics tracking
5. Loop through the simulation for the sim::total_time, again outputing data as before
Boiling it down, equilibration-time is primarily a convenience for resetting the underlying statistics tracking but will still be outputting values. If I remember correctly, it's quite an old piece of the code and recognised as being potentially a bit clunky/confusing.
From what I understand from what you've written, I think you're using the right split of equilibration and total-time-steps.
Hope that helps!
Luke