A cardholder-initiated transaction (CIT) is the card scheme term for what the main entry calls a customer initiated transaction (CIT): a payment that the cardholder takes part in at the moment it happens, for example by tapping a card in a shop, entering card details online or confirming a purchase with a stored card.
The distinction matters most for stored cards. Card network rules separate CITs from merchant-initiated transactions, which the merchant starts without the cardholder's active involvement under an earlier agreement, such as a subscription renewal. The first payment that sets up a stored card is normally a CIT, and in Europe it must usually pass strong customer authentication, while later MITs fall outside that requirement. Merchants flag each transaction with the right indicators so that issuers apply suitable risk checks.
"Cardholder-initiated" and "customer-initiated" mean the same thing. Visa and Mastercard documents tend to use "cardholder", while payment providers and much industry writing use "customer". The rules on storing card details are covered under stored credential.