Sorry for a bit of a basic question, but want to clarify things in my head. Is proper time quantified by the amount of physical process that an object, or physical system undergoes, for example the decay of an unstable element? And is the reason why proper time is independent of the frame of reference because such a physical process occurs without needing to introduce any coordinate system, thus regardless of the reference frame that you measure the process occurring in, it will occur at the same rate in the same manner?
For example, consider two observers equipped with identical clocks. One of the observers is considered at rest on Earth and the other is in a spaceship moving at a considerable fraction of the speed of light. Now is the notion of proper time that each observer, in their own reference frame will observe their clock to tick (cycle) at the same rate (however they will not observe each others clocks to tick at the same rate, as coordinate time is not frame-independent. Indeed from their own reference frame they will observe the clock in the other reference frame, moving relative to them, to tick at a different rate, as they are using their own coordinate time to measure the process). The point being is that if they both have identical clocks, then if the clock given to the observer on Earth ticks in a certain way for that observer in their rest frame on Earth, then the clock given to the observer in the fast moving spaceship will tick in exactly the same way in their frame of reference on the spacecraft (i.e. it won't suddenly change how it ticks to being aperiodic or non-linear, etc.)