Framed manifolds question

Question

Let a $\pi$-manifold be a manifold with the property that its normal bundle is trivial if it is embedded into $\mathbb R^n$ for large enough $n$.

Homotopy spheres are $\pi$-manifolds.

Here it is stated that the only spheres that admit a framing are $S^1, S^3, S^7$. Here it is stated that a framed manifold is a manifold with trivial normal bundle (therefore the same as a $\pi$-manifold).

These statements contradict each other. Which of the statements is wrong?

Answer 1

«Framing» means a trivialization of some vector bundle (tangent / stable tangent / normal for some embedding / stable normal). Naturally, existence of a framing depends on what bundle we're talking about.

Pages you're linking to explicitly state that

In one sense of the term, a framing of a manifold is a choice of trivialization of its tangent bundle

and

The term framed manifold (...) is most usually applied to an embedded manifold with a given trivialisation of the normal bundle