I was looking around on https://en.wikipedia.org/wiki/Homotopy_group, and saw that the definition of $\pi_n(X)$ is the set of homotopy classes of maps that map $S^n \to X$ (with fixed base points $a\in S^n$ and $b\in X$). Reading https://blogs.scientificamerican.com/roots-of-unity/higher-homotopy-groups-are-spooky, I see that even with $\pi_2(S^2)$, it begins to get difficult to understand (at least for me) what sort of "wrapping" we are doing; and of course $\pi_3(S^2)$ is even more confusing. But I had a question; what would happen if we change the definition to only consider homotopy classes of EMBEDDINGS $S^n \hookrightarrow X$? In this case, wouldn't the nastiness with higher homotopy groups go away, since $S^n$ can not embed in $S^i$ for $i<n$ (I don't know if this is true, but it is just a guess based on intuition)? I must admit I find this definition more intuitive regarding "detecting holes", or I guess in particular "spherical holes"; the official definition seems to me to be more about "interactions around holes" than the holes themselves.

As for group structure, maybe linear combinations, like what they have for cycles in homology groups (Intuition of the meaning of homology groups)? I have not studies these subjects in depth, but would appreciate any perspectives/references/pointers in a good direction. Basically, my question is this: it seems as if using an embedding $S^n \hookrightarrow X$ will make higher homotopy of spheres more in line with intuition (like what homology theory gives). Can this idea be made possible, with further details?