I've used the code sample described in the Adafruit lesson.
What happens after I connect the Uno to the USB cable is that the LCD lights up, but no text is displayed.
I've checked all connections and they look good. I am not 100%sure I have connected the potentiometer correctly, as this is the symptom I usually found online.
Could this also be a bad solder job? It's my first attempt and I guess it did not go that great.
Here are some more images, maybe someone with more experience can spot the error.
Update: here's what the board looks like once connected
and after removing the ethernet shield I get some weird flashes from time to time, only when I switch the potentiometer to maximum on one side, see video at https://i.imgur.com/ZclD2xz.mp4 .
Update 2: I realised that this is a 20-pin LCD screen and re-reading the datasheet and some online tutorials I switched digital wire connections to pins to the LCD 7-10 (DB0-DB3) - same result and then 11-14 (DB4-DB7) - garbled output, shown below. Still no idea what is wrong.