Core: Fix controllers resetting when starting input recording

200e26c added controller handling to MovieConfigLayerLoader's
LoadFromDTM but not SaveToDTM. This caused a regression. When starting a
new input recording, MovieManager::BeginRecordingInput calls SaveToDTM,
which doesn't set the controller values in the DTM header to anything.
Then it loads the newly created movie config layer, causing LoadFromDTM
to be called, which in turn causes zeroed out controller values to be
loaded from the DTM header. This leaves the player with all controllers
set to None, making it impossible to control the game.

To fix this, let's move the saving of controllers from
MovieManager::SaveRecording to ConfigLoaders::SaveToDTM.

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