Standardize the Extension Algorithms

[avik-pal]

Checklist

• [x] Standardized Kwargs
• [x] Forward the tracing kwargs and deprecate the ones in the Solver Kwargs
• [x] Use Jacobian Handling via jacobian!! in FastLM

[codecov[bot]]

Codecov Report

Attention: 45 lines in your changes are missing coverage. Please review.

Comparison is base (fa16796) 88.04% compared to head (bfaf014) 87.47%.

Files	Patch %	Lines
src/function_wrappers.jl	72.34%	26 Missing :warning:
ext/NonlinearSolveSIAMFANLEquationsExt.jl	72.91%	13 Missing :warning:
src/extension_algs.jl	76.19%	5 Missing :warning:
ext/NonlinearSolveNLsolveExt.jl	92.30%	1 Missing :warning:

Additional details and impacted files

```diff @@ Coverage Diff @@ ## master #341 +/- ## ========================================== - Coverage 88.04% 87.47% -0.57% ========================================== Files 27 28 +1 Lines 2174 2172 -2 ========================================== - Hits 1914 1900 -14 - Misses 260 272 +12 ```

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[ChrisRackauckas]

Are the 23 test problem failures known?

[ChrisRackauckas]

I presume that's what was fixed in the problem library, so it should have passed?