You're wrong. Koine Greek has the definite article (ho, hé, to). Bare nouns like "en arché" often express a definite temporal idea without an article, so English naturally supplies “the.” John 1:1 does use the article with logos (ho logos) marking a particular, identified subject. The grammar and context support reading the Logos as the definite divine agent, not an indefinite “a Logos.”
And so is "the" beginning, in English. Saying "at a beginning was the Word" is at least unnatural in English, if not incorrect.