As far as I can see, this word can be translated in both ways, e.g.:
The word “детермінатив” is more used. It is even listed in some dictionaries, e.g. Є. І. Гороть, Л. М. Коцюк, Л. К. Малімон, А. Б. Павлюк “Великий англо-український словник” (2011):
determinative [dɪˈtɜ:mɪnɘtɪv] п
- грам. визначальне слово; детермінатив.
(other translations listed here also suggest “визначальне слово” and “детермінативне слово”, however I'm not sure it's exactly the same meaning). The word “визначник” is used more rarely, but it is a literal translation of the word “determiner”, so it also should be understood.
The thing is that as far as I know the term itself is used not so often in Ukrainian linguistics, at least at average levels of Ukrainian linguistics (e.g. those taught at school). For example most of demonstrative determiners, distributive determiners, interrogative determiners and possessive determiners are classified as вказівні займенники (demonstrative pronouns), означальні займенники (“defining” pronouns), питальні займенники (interrogative pronouns) and присвійні займенники (possessive pronouns) for wide audience (and most of dictionaries also classify them in this way).