Both engagement and engagement mean engagement, and there are differences between them.

At first, people only used the phrase be engaged in sth. With the development of language, the combination be engaged in sth. appeared later.

The prototype of the engagement verb is engage, which means "employment" (= employment sb). Do a granular job);

So do something. It is equivalent to being "occupied" or "busy doing something", for example, he is writing one of his new books;

In general, it means to embark on a new difficult task. New or difficult) He is about to start a new career.