This is all correct except that financial accounting deals with external reporting and managerial accounting deals with internal reporting. But the distinction between bookkeeping and accounting is spot on IMO.
Pretty much, yeah. Managerial 101 builds on introductory financial reporting. The biggest difference between them are the intended users. While financial accounting is all about accurately stating your financial position for tax authorities, capital markets, banks, and other potential users, managerial accounting is internal. It's for managers and decision makers to give them the most accurate information possible. Because of this, there is no consistent standard used in managerial accounting like there is in financial accounting (GAAP or IFRS), but its largely similar I believe between different companies.