That's the part that is missing from it - it has the seller's name/business name/address, and the same for the buyer, and it states the item being purchased and the amount being charged, and it has 'Invoice [number]', and (if required) you can have VAT information on there too.
I have to admit part of this is that I have been spooked a bit by reading some resources that appear to say (in places) that invoicing is mandatory in Europe for all business transactions (even if you're not VAT registered). For example this from the ec.europa website (sorry had to truncate the link because it wouldn't let me post otherwise):
taxation_customs/business/vat/eu-vat-rules-topic/vat-invoicing-rules_en
Which initially appears to be about VAT, but the statements made after seem broader in scope. On the other hand, I've read some resources that say you don't need to invoice if you're not VAT registered. Because of the doubt, I'm considering a 'cover all bases' approach.