By the bye is not a phrase. The other two have a similar meaning but "by the way" is the most popular.

there are different

Probably not but please note that 'by the way' is MUCH more common (in England at least) than the other two. I have never in my (long) life wriiten 'by the by' or 'by the bye' so the problem of spelling did not exist for me. I have never spoken the words either! However, I am aware of some others using them.

Bye is used as goodbye and by is used as saying something is next to something for example the tree is by the lake

The first two have the same meaning, but 'by the way' is much more commonly used. The last is incorrect as others have said because 'bye' is used as a short form of 'goodbye'. 'Bye' is also used when talking about team sports competitions, when a team does not have to play a match but just moves on to the next round in the competition. For example 'My team has a bye this week, so I'm having a rest day.'