Which moves first

Which moves first? Is it NIFTY50 index moves first and then the NIFTY50 stock follows OR the NIFTY50 stock moves first and then NIFTY50 index follows?

Nifty50 is just an index based on it’s constituent stocks. Stocks move first and is mirrored by nifty or any index.

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Does same applies for BANK nifty

For any spot index the same.

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Since NIFTY50 reflects the mood of the overall market and its trend, doesn’t it influence the move of one NIFTY50 stock based on its trend?

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I think so. Also nifty futures should affect nifty which are independent of nifty stocks.