That doesn't look contradictory. An expression is a (syntax) tree, and every variable use has a unique path from the root to that use. This path crosses some sequence of binders, which we can label with integers.

If you assign integers from the inside out, as DB indices do, then the `x` usage in `\x -> \y -> x` passes one binder (`\y`) before landing on the correct one (`\x`), yielding `\_ -> \_ -> 1`.

If you assign integers from the outside in, as DB levels do, then the `x` usage lands on `\x` instantly, yielding `\_ -> \_ -> 0`.

De Bruijn came up with both systems, and it can be confusing to keep track of which one is which; but they are different systems.