re‧ceipt /rɪˈsiːt/ ●●● S2 noun
1 [countable] a piece of paper that you are given which shows that you have paid for something
Keep your receipt in case you want to bring it back.
in‧voice1 /ˈɪnvɔɪs/ ●○○ noun [countable]
a list of goods that have been supplied or work that has been done, showing how much you owe for them
According to my study, it seems that "a receipt" lists things that you did pay for already, and "an invoice" lists things that you have not paid for and must pay for.
However, Cambridge Dictionary says
invoice noun [ C ] US /ˈɪn·vɔɪs/ a statement listing goods or services provided and their prices, used in business as a record of sale:
You need to have a copy of your original invoice if you want a refund.
So, I reckon sometimes "a receipt" and "an invoice" are the same.
Is that correct?