A chicken burger involves a patty made of ground chicken, with no breading, just as a case where you had uncut turkey breast that was breaded and deep fried would not be a turkey burger.
See also what makes a salmon burger (ground and shaped into round patty) vs. salmon sandwich (whole fillet, may or may not be breaded).
If you have a slab of steak between buns, it's not a steak burger, despite still using beef.
You're saying these things like they're facts that answer my whole burger-sandwich question. But they're not facts - they're your personal understanding of each of those terms. My understanding of each of these terms is very different:
A "chicken burger" definitely includes most burgers at KFC, including chicken tender burgers, chicken fillet burgers, etc
If I saw "salmon burger" on a menu I would assume it would be a battered or crumbed salmon fillet on a burger, like most seafood burgers where I'm from in Australia.
If you asked me to picture a "steak burger" I would picture sliced strips of steak on a burger bun.
(if you were intending to help by offering your personal perspective as a data point, it would have helped if you included the region/variety of english you speak)
This is why the question is so difficult. People's understanding of the term "burger" (or "sandwich" for that matter) results in different interpretations and since there is no single trait that ties these interpretations together, there can be no hardline "definition" for the word "burger". This is true for every (natural) word in every language (technical terms excepted since their definitions inform the use of the word, not the other way around - e.g. scientific term "bug" vs everyday use "bug").
Dictionaries are useful, but they are also futile attempts at describing the infinitely nuanced interpretations of the signs used in language. The problem with the whole "is a hotdog a sandwich" bullshit is that the people who bring up that sort of thing see dictionaries and definitions as prescribing the meaning of the word, not describing it, which is a problem because that's not the intended use of a dictionary, and also because the written definition for any given word couldn't ever possibly hope to describe its meaning completely.
