Both these meanings for παῖς (child or servant) are correct. But the meaning of "lover" is not accurate.
Παῖς did, however, also mean the young lad or young girl, virgin (so maybe this is why it might has been related to lover's meaning, but παῖς is not the lover).
In modern Greek, the word in use is παιδί and its meaning is "child, kid". A really obsolete meaning is "boy" or "son", as opposed to girl or daughter.
Mainly in plural, παιδιά, except for children, it also means "guys", as a familiar address between children or adults.
example: παιδιά, πάμε μια βόλτα = Guys, shall we go for a walk?
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Vasiliki Baskou, Instructor/Director, https://learn-greek-online.com.