[Profile creation] Split Dice-specific signin logic from the generic one

This CL splits the Dice-specific sign-in logic (now called
ProfilePickerDiceSignInProvider) from the Dice-agnostic signed-in
profile creation logic (now called ProfilePickerSignedInFlowController).
This unblocks reusing the generic part of the code for lacros. It is
(almost) pure refactoring, with no noticeable behavioral changes (that
only requires minor changes in tests).

To simplify reviewing, both new classes are kept in the same .h and .cc
files. They will be split into separate units in a trivial follow-up.

Bug: 1227029
Change-Id: Ic6e0df8f7c3543c69fbd872cc52e9f57c8a0c72a
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/3107008


Commit-Queue: Jan Krcal <jkrcal@chromium.org>
Reviewed-by: Stefan Kuhne <skuhne@chromium.org>
Reviewed-by: Kush Sinha <sinhak@chromium.org>
Cr-Commit-Position: refs/heads/main@{#915904}