Have you ever used abstraction for hiding complexity?

In object-oriented programming, abstraction is often defined as “hiding internal complexity and exposing only essential features.”

But in real-world software development, we mostly use abstraction to:

• Define a common base class or interface, and

• Allow different implementations to be used interchangeably via that common contract.

For example, I might have a PaymentProcessor interface that both StripeProcessor and PayPalProcessor implement — so my code can depend on the abstract PaymentProcessor, not the specific implementation.

But here, I’m not really “hiding complexity” in the sense that the main goal is not to simplify the interface for an end user — it’s to make the system extensible and loosely coupled.

So my question is:

Shouldn’t we define abstraction more in terms of creating common interfaces for interchangeable behavior, rather than just “hiding complexity”?

Or — is “hiding complexity” still the right definition, but just a broader one that includes these use cases?