I believe you give a physical signature and its compared to what's on the back of the card.

It is never, ever compared in the US. In fact, I haven't signed the back of a credit card in years, and the only time I did was when I was in Europe where every cashier thinks they're a forensic handwriting specialist.

Not in the US.

For most terminals the small screen is actually a touch screen that (in the US) has a stylus attached to sign with.

However as you would expect, the signature is completely worthless and basically everyone signs with a simple scribble.

I draw little pictures for the cashiers at a store I frequent.

Most likely this. I'm not sure why card readers in Germany don't ask for the PIN every time but when they don't you're generally asked to sign the cashier's copy of the receipt and they'll verify the signature against what's on the card.

However not only will the cashier generally let you simply put the card next to the receipt while you sign (because this makes verification easier for them) but it doesn't seem like they actually apply any scrutiny: the signature on my card is very different from the one I use for signing these days and it never raised an eyebrow.

FWIW, I've signed for card transactions in all kinds of places across Germany, from small shops to large hotels. I can't figure out what triggers the decision between PIN and signature but I swear I've used either in the same place at different times for equivalent amounts.

The transactions with PIN are more expensive for merchants, but the merchant isn’t liable. They are done via the EC network.

The transactions with signature have lower fees, but the merchant is liable, and you’re actually authorizing them to do a Bankeinzug via the Elektronisches Lastschriftverfahren.

EC transactions are done instantly, ELV are done overnight.

Yes. Above a dollar amount depending on the store. But basically no one compares it the card and inputting it on the pad is so crude you might as well just write an X (which my signature isn't much more sophisticated than anyway these days).