Deputy or Vice? Let's deal first with the other meanings of vice as a noun, which are neresť and zverák (from different origins). Vice as in Vice President is from the Latin for “in the place of”, meaning somebody who has the authority to act instead of their superior (nadriadený/á), but without the idea of being sent somewhere else. In fact the use of Deputy or Vice (or Vice-) just depends on custom or convention, the typical combinations of words which people have learned and simply use out of habit. Maybe Vice sounds slightly more formal than Deputy, or it has less sense of subordination in it, so it is preferred in connection with President, whereas Deputy is more typical in connection with Sheriff.
Tricky Words in this week‘s OVI
Deputy. The original idea is someone who is sent off by somebody to speak or act for them in another place, so it is a calque of the Slovak poslanec. The French député (pronounced /depite/) has the same meaning, also used specifically for members of parliament, who are “sent” there by the voters who elect them to represent their (i.e. the voters') interests. In English of course another idea has got mixed up in the word, which is that the deputy is subordinate (podriadený – another calque) to the person they represent, so this term is not typically used in English for political representatives. Maybe if it was, it would remind them more often that they are supposed to speak and act on their voters' behalf (literally konať nápomocne).