that's true, if all business is done within EU and both parties have EU VAT ID, then the seller does not charge tax on the invoice and adds something like "it is the buyer's responsibility to pay VAT according to the tax laws in the buyer's country"

On the buyer's side, the buyer needs to pay additional VAT in their home country just as if they got the product locally.

and yes, the VAT ID is important because if you don't get that on the EU invoice you can't actually use it for anything other than just a proof of purchase.