The intent is when in story mode to prevent players from attempting to heal multiple times. Just like when in story mode and usig actions and trying to use the standard recharge mechanic for actions does not work, and so are limited to once per 'encounter' or act.
Therefore, yes, technically a first aid check during combat would count as the single check for the act. Some GMs house rule allowing a new check if the PC sustains additional damage after the first aid check has already been made.
The 'exception' is a check for long-term care, which influences the nightly recovery check each player can make.
I don't have the book in front of me, so I'm going off the top of my head. (ie this might be slightly off).
Essentially, every night of rest each PC may make a resilience test to recover either Wounds or Criticals. The difficulty is based on how wounded they are. 1d if they are Wounded (no criticals) and 2d if they are critically wounded (1 or more criticals). The number of successes rolled on the resilience test is either the number of Wounds recovered or the severity value of the critical(s) that can be removed. This can be modified by the environmental conditions. For example, resting outdoors would likely provide no bonus, resting in a poor quality inn room might provide 1 fortune die, resting in a hospice of Shallya might provide 3 fortune dice, etc.
Successful long-term care checks provide additional fortune dice to the PC's resilience test, thus allowing the potential for more successes to be rolled.
As a GM, I require at least 6 hours of "restful" sleep/inactivity, within an 8 hour period (or so), before allowing a PC to make a resilience roll. I.e., a PC that doesn't sleep, or only sleeps for a couple of hours a day, etc, will not be able to make a resilience test to recover wounds that day.