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I guess I should put a TL;DR somewhere. But to answer your question, "bara" doesn't have a clear meaning but usually refers to drawn sexualized depictions of traditionally-masculine men. Yes, it's often used to describe what type of story it is, usually indicating that it sexualizes men who are muscular or have non-mainstream bodies. 

To explain the difference between bara and yaoi, I'd have to explain what yaoi is briefly. Essentially it is a genre of manga and anime that focuses on a male same-sex relationship that's typically made by women for women. The term then began to be used for any story that focused on a male same-sex relationship even if it didn't come from Japan. However, due to marketing and audience, stories that are usually labeled as "yaoi" or "BL" (which more or less mean the same thing except "yaoi" usually depicts more sexual stories) usually contain pretty-looking, young men as the focus. There have been "yaoi" that includes more muscular characters or older characters, but they're pretty rare. The lack of a clear definition of "yaoi" and "BL" is actually kind of a problem for some Manga and Anime such as Yuri On Ice!!! and Ao no Flag that features gay characters (although some people argue that Yuri on Ice!!! just implies it which I'm not getting into) and people call them "yaoi" when they don't fit the genre. "yaoi" is definitely not the same as "bara" but which is which depends on what the artist calls it really. Some say their work is "yaoi", some say it's "bara", and some say it's both. But generally, I'd recommend not calling a gay VN that has slender characters "bara" or a gay comic that has musclechub characters "yaoi". People have expectations when it comes to these terms so not caring about how you use these terms shows that you are very ignorant about these terms. It's like calling Khloe Kardashian "goth" because she wore black one time or a cute anime girl "a tomboy" because she has short hair. Yeah, these terms don't have exact definitions but that's no excuse for using these words for whatever.

Essentially, there are no clear definitions of "bara" and "yaoi" but most people expect muscular/beefy, sexualized gay men with "bara" and slender, pretty men with "yaoi". Whether or not something is bara, yaoi, or both depends on the creator.

As for the difference between "bear" and "bara", "bear" refers to a specific subgenre of the gay community or a gay man who's hairy. "bara" usually refers to fictional men since the proportions of the bodies tend to be really exaggerated. However "bara" can be used to describe real-life men if they are particularly buff or beefy, though I personally don't do this. Basically, there is a lot of overlap with the two terms, but not all bears are "bara" as not all bears are muscular, and not all bara feature "bears" since not all bara depict hairy men.