Move data handlers into constructor

Previously, because the data handlers were external to fresh-data and typically props of a provider component, `setDataHandlers` was called whenever the props changed. However, since the ApiClient is basically re-created in most of those cases now, it makes sense to just include these data handlers in the constructor of ApiClient, etc. This came up in a PR review comment: https://github.com/Automattic/fresh-data/pull/202#pullrequestreview-259694818

Start by tracing ApiClient construction and the current setDataHandlers calls from provider components, using the linked PR review for context. Move the data handlers into the ApiClient and related constructors, then verify that prop changes no longer require separate handler updates.

Written by the indexing model from the issue text.