Learn Hardcore Indonesian: Dia mengajak saya belajar di perpustakaan. - He/She invites me to study at the library.

In Dia mengajak saya belajar di perpustakaan, dia just means “he/she”.
To know whether it’s he or she, you need extra context from the conversation (for example, if people were already talking about a man or a woman). The sentence alone doesn’t tell you.

What is the exact nuance of mengajak here? Is it “invite”, “ask”, or “persuade”?

Mengajak basically means “to invite / to ask someone to do something together”. The idea of togetherness is important.

In Dia mengajak saya belajar di perpustakaan:

It suggests “He/She invited me to study at the library (with him/her)”
It can also sound like “He/She asked me to go study at the library”
It usually does not mean a formal, ceremonial invitation (for that, mengundang is used).

So:

mengajak → casual, “Come on, let’s … (together)”
mengundang → formal invitation (weddings, official events, etc.)
meminta → to request/ask (not necessarily together): Dia meminta saya belajar = “He/She asked me to study.”

Why is it mengajak saya belajar and not ? Is required?