Despite what these self-posessed "experts" have to say, spelling is largely a matter of context. There are in excessive of over 500 results for "amtrack" on this site alone, which in itself lends credibility to the theory of "relativity" at least in terms of spelling (we're not talking Einstein, folks!). A little research reveals that in some cases it is in fact spelled with the "ck" namely when you are referring specifically to the tracks themselves of the rail network. However, when discussing the government-run agency in itself then the "c" is "removed." This at least is my understanding. Source: My former neighbor Ed used to work for the rails.
Spelling is not a matter of context, and it has nothing to do with relativity. Spelling is concrete. Just because some people spell a word wrong, that does not mean it's okay to spell a word that way.
The name of the corporation is Amtrak; the correct spelling is Amtrak. End of story.
If a few people spelled your name "Brehnda" instead of "Brenda", would you be okay with them using "Brehnda"? Of course you wouldn't.