I think you're conflating soft/hard real time with criticality (although they're often aligned)
Hard real time refers to a total system failure, even if it has trivial consequences. For example if a printer has a timing error at any point the page it's printing is irrecoverably ruined. It's hard real-time but low criticality.
Soft real time is where you can miss deadlines but the performance / experience degrades rather than fails. e.g. frame rate drops in a video game. The frames are still valid when delivered late, but the experience is poor.
> Slight nit, real time systems are designed for a maximum worst case latency within some spec, not minimum latency.
Indeed, in fact at times you don't even want a substantially lower minimum latency because it may cause unreasonable jitter in your output.
And in the context of real time systems, a missed deadline is either equivalent to a catastrophic error (hard real time) or really terrible but non catastrophic error (soft real time). Soft real time applications like audio (no one dies because of a buffer underrun) can sacrifice the maximum worst case for better average case, because the worst case depends on load and can be worked around. Hard real time cares a lot more about the maximum worst case, like say a CNC machine where missing a deadline can cause a mechanical failure and break the part.