I do not have an answer and admit that I have not yet fully read in detail all answers and comments in this question but it srikes me that <PERSON>'s principle of which <PERSON> was a great admirer was not yet mentioned.
In short: <PERSON> was of the opinion that only when other masses in the universe (fixed stars) exist a moving accelerometer can ever measure a force.
To us today this seems like a stark statement. But please consider this:
<PERSON> writes on pp.3-4 of his book on classical mechanics:
"<PERSON>'s principle of relativity:
There exist coordinate systems (called inertial) possessing the following two properties. 1. All laws of nature time at all moments of time are the same in all inertial coordinate systems. 2. All coordinate systems in uniform rectilinear motion with respect to an inertial one are themselves inertial.
In other words, if a coordinate system attached to the earth is inertial, then an experimenter on a train which is moving uniformly in a straight line with respect to the earth cannot detect the motion of the train by experiments conducted entirely inside his car.
In reality, the coordinate system associated with the earth is only approximately inertial. Coordinate systems associated with the sun, the stars, etc. 562
are more nearly inertial."
It is <PERSON>'s last sentence that indicates the srong connection to <PERSON>'s ideas. I also take from this that <PERSON> did probably not want to try to give a formal definition of an inertial system since his formulation in 1. and 2. is obviously circular.
As I said: no satisfactory answer to the question what "is" an inertial system. It boils down to taking the frame that is most convenient to do calculations in full agreement with <PERSON> previous answer here.
One last remark: the above link <PERSON>'s principle brings up the paper
<PERSON>, A Theory of Inertia Based on Mach’s Principle.
It can be the starting point to find some more references, esp. from <PERSON> who wrote the book "The relativity of inertia" (1972). Unfortunately I have neither the English translation of that, nor the original German edition. 346