Thanks everyone and sorry I disappeared for a while. I should be working really!

Here is the recipient address I gave the sender.
This was in the Recieve Tab of my Electrum yesterday night. He wrote that it's the address he sent to, including the address, for confirmation.

1JW9NJneiHVqpvDgcNah4FLXRF62Mw1Np5

Sum in Bitcoin: 0.07592263

I double-checked that it's what I sent to the seller in the chat on Local Bitcoin Exchange. It is also what he said that he sent the payment to, via chat on Local Bitcoin Exchange. This guy gives every sign of being honest and decent and a chatted a bit with him while waiting for the payment. I just don't get scammer vibes from him at all, even though I realise anyone could be tricked. His high score on Local Bitcoin Exchange also indicates he's honest.

It's in fact worth more than I thought in EUR, it's EUR 611. I paid him in another currency.

Now I will read all the responses since my last comment, very carefully to understand it, then follow up.

In the meantime, I would be super grateful for help by an experienced person, to check the blockchain.

I understand what you are saying about the Transaction ID and I will ask him for it, if needed. It's still really early for him, due to time zones and it would be good to know the verdict of Blockchain first!

This is the first time I've had problems with a transfer, so from that perspective it's suspicous.