Revert "Revert "Forms: Finally store the feedback in the new format ""

[simison]

Revert of a revert PR when we're ready to attempt this again :-)

Reverts #44924

[github-actions]

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Code Coverage Summary

Coverage changed in 1 file.

File	Coverage	Δ%	Δ Uncovered
projects/packages/forms/src/contact-form/class-contact-form.php	572/1047 (54.63%)	-0.56%	0 💚

Full summary · PHP report · JS report