You are correct and that is what's missing from the Help article on that topic. Some providers do charge for that SMS text to the UK. Mine doesn't because it's flagged in the system as the iMessage activation number. I'm in Canada by the way.
In addition, it's done behind the scenes. Unlike a text where you have a record of a conversation between say you and me, you would not know of this until you see your bill.
If you get an error when trying to activate iMessage or FaceTime - Apple Support
Read this line very carefully:
Make sure that you’re connected to a cellular data or Wi-Fi network. If you're using an iPhone, you need SMS messaging to activate your phone number with iMessage and FaceTime. Depending on your carrier, you might be charged for this SMS.
That's why you were charged. If you want to dispute it with your providers, then go ahead and do it.