Can we define higher-order tangent spaces of a manifold with equivalence classes defined equating second derivatives?

No, in the case that $M\subset\Bbb R^n$ (or $\Bbb C^n$ or $\Bbb P^n$) is (locally) the image of a parametrization $\phi\colon U\to\Bbb R^n$, $U\subset R^k$ open, you need to look at the span of the first and all second partial derivatives of $\phi$. So, the second-order tangent bundle will generically be a bundle of rank $k+\frac12 k(k+1)$. (Of course, this "embedded" notion fails to be a bundle when the submanifold has unexpected shrinking of the osculating space.) The appropriate abstract construction is that of a jet bundle (in this case the $2$-jet). See, for example, Wiki, Hirsch's Differential Topology, and/or Guillemin/Golubitsky's Stable mappings and their singularities.