Gf impl code review suggestions

[vincentmacri]

Cherry-pick sagemath/pull/35609 and resolve all the conflicts. Also some minor suggestion.

This looks like a lot of changes but it's almost entirely changing impl to implementation in various places.

I'm running tests locally, and will unmark this as a draft once tests are passing.

[vincentmacri]

So far tests are passing locally. @adnanbhat7 I think you can approve the CI run here to make sure everything is working (although we'll still need to do that on the main repo as only some tests will work on your repo).

[adnanbhat7]

@vincentmacri Thanks for the review and for raising the PR! I've approved the CI .
I should not merge it until the failing checks are resolved.
I will merge your PR into my develop branch. After that, I will merge my develop branch into my fix_implementation branch and push the changes, which will then update the original PR in the organization repo.

[vincentmacri]

I should not merge it until the failing checks are resolved.

The documentation build failure is irrelevant, there's an existing issue with building the docs on CI. Basically we are hitting disk usage limits on the doc build. The build and test failure was spurious, I couldn't reproduce it locally. So I think this is ready to merge into your branch.

I will merge your PR into my develop branch. After that, I will merge my develop branch into my fix_implementation branch and push the changes, which will then update the original PR in the organization repo.

I made this PR against your fixes-GF_imple branch, so once you click merge here it should update your PR immediately, no need to mess around with your develop.