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Yes, indeed a LN invoice posted on public forums is not indicated.

In this particular case is just about onboarding new people and making an exsperiement.
As I said in the instructions post: you can use whatever LN wallet. For those "privacy" obsessed people, they could just spin-up a new LN wallet and use it exclusively for this experiment, I don't care.

Paying for a beer, I t don't think is such a danger to be arrested or killed or something. Sometimes people are going too extreme with all these "privacy" concerns.
You will find from a LN invoice who is receiving but not who is sending or from where...

how can we use LN for receiving payment in a private way?

I wrote several guides about that (I hope you read them):

.... and many more.

We should start living without fear anymore.

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