In-app provisioning is another name for push provisioning, the process by which a cardholder adds a card to a mobile wallet by tapping a button inside the issuer's own app instead of typing or photographing the card number in the wallet. The issuer's app passes the encrypted card details to the wallet, and a payment token is created for that device.
The two terms describe the same flow from different angles. Push provisioning refers to the direction of travel: the issuer pushes the card to the wallet. In-app provisioning refers to where the cardholder starts, and is the term used in the wallet providers' developer documentation. Some issuers reserve the phrase for the case where the wallet and the banking app sit on the same phone, and use push provisioning more broadly to include sending a card to a merchant's site or to a wearable.
In both cases the customer has already signed in to the bank's app, so the issuer can usually approve the request without a further identity check. The steps that follow are the same as for any card provisioning: the card network's token service provider issues a token under network tokenization and the card becomes usable in the digital wallet.