I think you're right OP, both phrases are about people keeping their options open.
But there is good and bad 'keeping options open'. Good is genuine uncertainty. Bad is flakiness or control.
'That should be ok' followed by checking and confirming, or being able to give a yes or no when you check, a few days later, is fine. It reflects real uncertainty. Being deliberately vague, so as to be able to trick you, or deny responsibility is obviously not ok. In that case no definite yes or no would ever be given.
So, if you really need something, on that date, you have to press for a yes / no answer and, if you don't get one, say 'I clearly can't rely on you, so forget I asked, I'll make alternative arrangements'.
In your boss's case though, I'd have taken 'that should be ok' as a yes, as I'd hear the emphasis on the 'ok' not on the 'should'. Unless it was firmly on the should, in which case I'd press for a yes / no, then or later.
Likewise 'I'll try' can mean they really will but cannot commit, or maybe if it suits them, or that they're giving themselves a get-out for not delivering. Whether you accept an 'I'll try' depends on the importance and urgency of the issue.
If you really needed say a prescription to be collected in your DH's 'lunchbreak town' and stressed the importance and urgency, would he commit to it? Would he do it? Willingness to prioritise others' needs over his own wants is the issue there.