Let's stick to SR here. Given an accelerometer, one can determine locally if one is moving inertially or not? Can one determine locally if a particular instrument is an accelerometer? If one has no accelerometer, then inertial movement has to be determined by absence of forces on the body. This can be done by knowing the full distribution of matter in the universe, and ascertaining that that distribution of matter does not exert any force on the particle, which seems to be a global determination?