Individuals CAN be VAT registered as sole traders, and hence if you were offering business coaching, they would be able to reclaim any VAT charged as input tax assuming you are also VAT registered.
As HMRC explains, there is a difference between a receipt and an invoice. A receipt is an acknowledgement of payment and is what you receive at Tesco having just paid them. An invoice is a request for payment regardless of any VAT element. The HMRC link even states “The VAT amount if applicable”.
If you are VAT registered and the sale is retail, then you can issue a “simplified invoice” for transactions under £250 which does not need to carry the customer details. In general, you’ll find that HMRC are happy with you doing the same if you are not VAT registered and simply miss out any reference to VAT such as VAT number or VAT value.
The main reason for adding the customer name and address is so that you are clearly stating WHO owes you for what and how much. A retail sale is one whereby the payment is taken at the point of transaction, so there is no need to state this.