Apparently the use of 'Ohiya' should be limited to the staff who are serving their customer at traditional eateries because it is actually a traditional code.
So, you can ask for a 'Omizu' then your sushi chef would serve a glass of water and may say "Here is your 'Ohiya', Sir." But apparently not the other way around - I didn't know this but the other way around somehow sounds wrong.
But if you must:
I'd only use 'Ohiya' at traditional Japanese eateries (hence "chic" as prev. comment), not in a French restaurant or McDonald's.
Also, if I were invited to my friends/families house, or they are serving food & drink as the host, I'd use 'Omizu'.
Given that 'Ohiya' is a code amongst catering staff, asking for an 'Ohiya' is the equivalent of calling your host "waiter/waitress" - sounds a bit pretentious.
Another thing is that 'Ohiya' has more subtle nuances than 'Omizu'.
For example, you can use 'Omizu' to mean 'bottled water' whereas 'Ohiya' can only be used when you mean a single portion of water in an open container like a glass/Yunomi, not in a pitcher or a bottle.
Anyways, unless you are working at a bustling sushi restaurant, you don't have to worry about it - it's like walking into a supermarket and asking for a 'code 300' which is an internal code for a 'manager'. If you are just a customer, you don't need to know such thing and if you use it, that won't impress anyone.
The only difference here is that you'd better to know what 'Ohiya' is, but that doesn't mean you have to use it.
'Ohiya' is a reserved (and quite archaic) code for the people who are serving you. Stick to 'Omizu' and you'll save yourself from minor embarrassments ;)