Cast INTEGER to BIT confusion

Hello PostgreSQL Community,

I have 25+ years of experience with some other RDBMS, but I am a PostgreSQL starter, so I assume the following is rather a simple beginner's question...:

I like to store just a single bit but that can be either 1 or 0, so I tried to do this:

CREATE TABLE T (c BIT);
INSERT INTO T VALUES (1);
-- I MUST NOT change both lines in any way as these are part of a third-party application!

Unfortunately this tells me:

column "c" is of type bit but expression is of type integer

So I logged in as a cluster admin and I tried this:

CREATE CAST (integer AS bit) WITH INOUT AS IMPLICIT;

Unfortunately that tells me:

cast from type integer to type bit already exists

This is confusing! Apparently PostgreSQL 15.3 comes with the needed cast out-of-the-box but it does not apply it? This is confusing!

What is my fault?

Thanks!
-Markus