Realtime is a fluid (AFAIK) definition. In general, realtime concerns reaction time to inputs - how much (sic) of inputs are processed "in time". On one hand, you may want the system to react "quickly" to 99% of inputs on a pin. On the other hand, you may not care (within reason) when the system responds to UDP packet. To recap, realtimeness concerns amount of events on the same input processed "in time" and number of inputs subject to timeliness. Hence general confusion and multiple definitions.

I would say there are two (2) sets of realtime definitions.

1) concerning amount of events

Hard realtime system always produces output to an input in predefined time (subject to clocking). This time can vary between different inputs, but is fixed for a given input.

Soft realtime systems are bound - output time is bounded to some number.

2) concerning number of inputs

Hard realtime system has bound reaction time to every possible input

Soft realtime system has bound reaction time to "critical" inputs.

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Definitions used in practice mix the two in some ways. The most prevalent definition I happen to come across in the general internet is something like "realtime system has bounded, low-variation reaction time to critical inputs".