Why is Pyrogram defined as both Client Library and Framework?

Simply because it falls in both categories, depending on how you use it.

Pyrogram as a client library makes it easy and intuitive accessing the Telegram API by offering idiomatic Python code

that is generated or hand-written. Low-level details and client-server communication protocols are handled under the

hood. Pyrogram acts as a client library when *you call* its methods and use its types in a batch application that

executes a set of instructions.

Pyrogram as a framework makes it easy to handle live events by allowing you to register event handlers that will be

executed as soon as they arrive from the server side. Pyrogram acts as a framework when it's Pyrogram itself that

*calls your code*, that is, your registered event handlers. Such applications are usually started and left online

indefinitely, until you decide to stop them.