If the GitHub repo has seen new commits pushed to it, while you were working locally, I would advice for:
git pull --rebase
git push
The full syntax is:
git pull --rebase origin master
git push origin master
That way, you would replay (the –rebase part) your local commits on top of the newly updated origin/master (or origin/yourBranch: git pull origin yourBranch).
See a more complete example in the chapter 6 Pull with rebase of the Git Pocket Book.
I would recommend a:
git push -u origin master
That would establish a tracking relationship between your local master branch and its upstream branch. After that, any future push for that branch can be done with a simple:
git push
Since the OP already reset and redone its commit on top of origin/master:
git reset --mixed origin/master
git add .
git commit -m "This is a new commit for what I originally planned to be amended"
git push origin master
There is no need to pull --rebase.
Note: git reset --mixed origin/master can also be written git reset origin/master, since the --mixed option is the default one when using git reset.