In modern usage "Geschäftsführer" would be the most usual choice.
"Direktor" is more typical for a 19th or early 20th century business environment, as in "der Herr Fabrikdirektor", or for some educational institutions (Museumsdirektor, Gymnasialdirektor, Volkshochschuldirektor) etc. You also find it in Zirkusdirektor which again comes from the old days.
Being somehow old-style, in an industrial context the word "Direktor" is likely to evoke in people's minds the image of a fat avuncular type in striped trousers, black tailcoat, and a tophat on his head and of course smoking a cigar. Don't forget the monocle. So that's a 1870-1920 stereotype.
The "Geschäfts" part of "Geschäftsführer" bears no implication of the business he/she manages being a "Geschäft" in the sense of "shop". "Geschäftsführer is simply "he who runs the business". So you may well be the Geschäftsführer of a car repair company or Geschäftsführer of a business consultancy or whatever.
Interestingly, however, "Ladenhüter" is not "he who looks after the shop" (which would be a reasonable literal understanding), it rather is stuff on sale that sits in the rafters and does not get sold for months.
So, to answer your conclusive question: yes, you may use "Geschäftsführer" as a synonym of "CEO", or vice versa. All those who want to be modern, dynamic, startupish, cool and internationally recognizable nowadays call themselves "CEO" (notably also those who run single-person businesses). But as long as you operate in the German speaking part of the world, or want to avoid hip terminology, "Geschäftsführer" is a solid solution.