Align APIs with new `_mut` conventions

As per [RFC 52](https://github.com/rust-lang/rfcs/blob/master/active/0052-ownership-variants.md), use `_mut` suffixes to mark mutable variants, and `into_iter` for moving iterators. Additional details and motivation in the RFC.

Note that the iterator *type* names are not changed by this RFC; those are awaiting a separate RFC for standardization.

Closes #13660
Closes #16810

[breaking-change]