I have developed a website with a friend of mine, and the site is working. But it is not currently not in a form which is presentable to the public, which will take us a little bit longer to do. We both believe that the idea we have come up with is probably worth trying to get a patent for. I was just wondering if we need to have some kind of evidence that we are the first people to have developed our idea, in the form of a video or something similar, or if for example the code we have stored on bitbucket would be an appropriate way of demonstrating this in future, if needs arise? In other words, is it when the idea is presented to the public first which allows you to patent, or just proof that you developed the idea first? Thanks.
For patent do you have to make your discovery public?
The answer to your headline question is yes. The central societal trade off in the patent system is you teach us all about your invention and we (the government) give you a possibility of a time limited monopoly.
But, the body of your question is not actually that question. The U.S., like most of the rest of the world, is a "first-to-file" country. One just files an application and you get that date as a priority date. If you know of anything that preceded you which tends to show that you might not be first or that might be combined with other information to argue that your invention is obvious, you will need to let the USPTO know.
The information you are talking about can still be useful if someone later claims that you did not invent but specifically took the idea from them.
If the invention is already visible to the public you will need to file soon, the U.S. has a modified one-year grace period to file following public exposure.