Three words, and only two distinct documents. Invoice and bill generally name the same thing from opposite sides of a transaction: the supplier issues an invoice, the buyer receives what their software may call a bill. A receipt is genuinely different, because it is issued after payment and confirms settlement rather than requesting it.
Invoice and bill: one document, two viewpoints
The seller raises an invoice and records a receivable. The buyer records the same document as a bill or a payable. Some accounting packages use the words to distinguish incoming from outgoing, which is a helpful convention rather than a definition, and other packages reverse it. There is no reliable technical difference.
Receipt: the other end of the transaction
Issued after payment, confirming that a specific amount was received against a specific obligation. It proves settlement. In business payables the remittance advice does much of this work from the payer's side, telling the supplier which invoices a payment covered so they can allocate it correctly.
The one that catches people out
A statement, which is none of the three. It lists what a supplier believes is outstanding across several invoices. It is a reconciliation tool and should never be paid from, because paying from a statement means paying without the underlying invoice and without any match. That mistake produces duplicate payments reliably.
Questions people ask about invoice vs bill vs receipt
Which word should we use internally?
Whichever your system uses, consistently. Two vocabularies means people searching the wrong list.
Can a receipt substitute for an invoice?
For record-keeping purposes requirements vary by jurisdiction and by what the document contains. Check your own obligations.
Why should we never pay from a statement?
Because it carries no line detail to match against, and it may include invoices you have already paid or never received.