I don’t think that I’ve explained myself well enough.
I assume your circumstances are that your Owners Corporation (O/C) is working off one of the Model By-Laws covering the keeping of animals on a Lot or on the Common Property, where animals / pets cannot be kept there without the prior written consent of the O/C.
It’s relevant in your circumstances that a consent of the O/C can, and usually does, include conditions that therefore form part of that consent.
You stated in your post (#1) that in your 4 lot strata, 2 owners are prepared to vote for consent on the basis of a included condition that the pet (a dog) must be kept on a leash when on the Common Property, and the other 2 owners, including the one seeking consent, will also vote for consent but without the included condition.
So if the Motion is for the consent to be granted with the included condition, you and the like-minded owner will vote in favour and the other two owners will vote against, so it’s a tied vote and that Motion is lost; no consent and no dog!
Conversely if the Motion is for consent with no on-leash condition, the other two owners will vote in favour and you and the like-minded owner will vote against, so again it’s a tied vote and the Motion is lost; no consent and no dog!
There’s a requirement for O/C’s to behave reasonably in such circumstances, so the owner seeking consent could take the matter to arbitration, where I’m sure that a preference by 50% of owners not to refuse consent, but rather to grant it subject to a very reasonable condition (in my opinion) would be a upheld.
I hope that makes the position clearer.