Page split where existing rows are moved to different page occurs only for updates and non-sequential inserts?

Suppose page is full, and an item is inserted in order, then that item will get added to the next page. There will never be page split such that some rows are placed on a page and some rows move to another page. Where as if an update is performed such that an item needs to be placed somewhere in an already full page, then a page split happens where some rows are kept of this page and other rows are moved to a new page. So a page split where **existing** rows are moved to different page occurs only for updates and non-sequential inserts?