While I endorse Andreas Blass's answer, a different perspective is the following which I think cuts more to the heart of the issue.
First, there are many quite distinct formal logics. That classical logic works some way doesn't mean every logic has to work that way. This means there's a plurality of possible options, and we can choose which seems to work best for our purposes. With regards to modeling informal reasoning, this is akin to having multiple physical theories (where physical theories model reality). To this end the answer to your question "what dictates that the OR which is defined the way it is now, is the correct one?" is "nothing" and different logics do make different choices. However, making a different choice is making a different logic, not changing what The One And Only Logic is.
To this end, different logics correspond to different ways of modeling informal reasoning. Just like we may use Newtonian Mechanics rather than General Relativity, say, logics that may not perfectly or completely capture every aspect of informal reasoning may nevertheless be good enough for our purposes and much more convenient than a more precise account.
In a different direction using the same distinction, nowadays we kind of make fun of Aristotle's physics. We tend to have the idea that he needed to go outside and do some real experiments. However, many of Aristotle's ideas do seem to fit observations better. Newton's ideas seem obviously wrong. In day-to-day experience, an object in motion does not remaining in motion. Of course, nowadays we know that applying Newton's ideas to day-to-day experience requires a much more in-depth modeling. Likewise, modeling a piece of informal reasoning in a formal logic may be (is) a much more involved and subtle exercise than a string replace that replaces "or" with "$\lor$" and so forth. Natural language is not entirely compositional or context-free.
Finally, informal reasoning in practice is filled with flaws. These mathematical models are also idealizations, so at times where they depart from informal reasoning, this may be to their benefit.
In a totally different vein, formal logics are mathematical structures and logicians and mathematicians have plenty of other reasons to study them regardless of how well they correspond to informal reasoning.