Car has power, but no crank/start

Hello

I have a 2005 C180 Kompressor (I'm in Europe). Soon after putting on a new car battery, I got a message on the dash warning me about the alternator. I took it to a repair shop and they did some measurements, showing that the alternator was ok (generating 14,1 A) so I kept using the car with that warning on.

Today, after a longer drive, as soon as I stopped the engine I wanted to turn it back on (remembered I had to go somewhere), but nothing happened. Dash lights turned on, but I heard no cranking. I checked the starter engine fuse and it's okay, the battery seems to have enough juice (lights are not dimmed, audio system is loud, etc.)... I just hear no cranking.

After opening the hood, I can hear something like a relay click, especially when I turn the key from position 3 to position 2.

Any thoughts on what I should do/check next, apart from having it taken to a repair shop?

Thanks