manifolds – An intuitive explanation of Tangent space

Given a manifold M, I imagine the tangent space at p, $T_p (M)$ as follows:

  1. Consider a higher dimensional vector H space where M is embedded.
  2. Look at all smooth curves in M that pass through p.
  3. Compute the velocity vector of the curve at p. These vectors will lie in higher dimensional vector space H.
  4. Together these vectors form the tangent space.

The formal definition however is as follows:

For any given smooth curve $lambda$ in M passing through p, define $v_{p, lambda}$ as a linear map that maps smooth functionals on M to a real number as follows:
$$v_{p, lambda}(f) = (fcirc lambda)'(0)$$
where $lambda(0) = p$.
The set of all such linear maps form the tangent space.

How do I reconcile the intuitive picture with the actual definition?