Some people don't care about proof; they just fill in the blanks with whatever they find and call it good.
If you want proof, if you want to do it right, you still have to do it the old-fashioned way, with collecting the documents that show the relationships and such.
I know I have stuff in my trees that isn't correct; bit by bit I go back and track down what documentation I can to verify what I have. But you can't trust that others are doing that. Especially with the amount of "information" floating around these days on Ancestry. It's all too easy for people to just hit the button and add stuff with no attention to whether or not it makes sense.
In short, how to prove "this is grandma' with the correct details: don't blindly copy stuff that doesn't have documentation. Look for things like census records, death certificates, birth certificates, marriage licenses, obituaries or other newspaper articles, whatever you can find. The bad news is that sometimes those won't match, either, and all you can do is try to judge the value of each source independently and go with what seems to you to be the best information. People are sloppy and the only 'defense' you have against that is not to be sloppy yourself. It is a slow and painstaking process and the internet is both a blessing and a curse to the modern genealogist.
*DON"T* get hung up on the spellings of names--the farther back you go, the less likely there was one "correct" spelling. And names do 'morph' over time, Bashor being a case in point. When the first members of the family arrived here, it was Bosshaar or something like that, and other variations of the name occured along the way. The family named "Cook" now might have been "Koch" a few generations back. Names got changed at Ellis Island, names got creatively spelled by barely literate officials filling out paperwork along the way, and the people who had the names were frequently not literate at all. My great-great-grandfather was William Britain, Brittain, Briton...you get the idea. If you get hung up on the "right" spelling, you *will* overlook things that you need.