Unix mail programs have had two approaches to handling your mail
Historically, Unix mail programs (what we call 'mail clients' or 'mail user agents' today) have had two different approaches to handling your email, what I'll call the shared approach and the exclusive approach, with the shared approach being the dominant one. To explain the shared approach, I have to back up to talk about what Unix mail transfer agents (MTAs) traditionally did. When a Unix MTA delivered email to you, at first it delivered email into a single file in a specific location (such as '/usr/spool/mail/<login>') in a specific format, initially mbox; even then, this could be called your 'inbox'. Later, when the maildir mailbox format became popular, some MTAs gained the ability to deliver to maildir format inboxes.
(There have been a number of Unix mail spool formats over the years, which I'm not going to try to get into here.)
A 'shared' style mail program worked directly with your inbox in whatever format it was in and whatever location it was in. This is how the V7 'mail' program worked, for example. Naturally these programs didn't have to work on your inbox; you could generally point them at another mailbox in the same format. I call this style 'shared' because you could use any number of different mail programs (mail clients) on your mailboxes, providing that they all understood the format and also provided that all of them agreed on how to lock your mailbox against modifications, including against your system's MTA delivering new email right at the point where your mail program was, for example, trying to delete some.
(Locking issues are one of the things that maildir was designed to help with.)
An 'exclusive' style mail program (or system) was designed to own your email itself, rather than try to share your system mailbox. Of course it had to access your system mailbox a bit to get at your email, but broadly the only thing an exclusive mail program did with your inbox was pull all your new email out of it, write it into the program's own storage format and system, and then usually empty out your system inbox. I call this style 'exclusive' because you generally couldn't hop back and forth between mail programs (mail clients) and would be mostly stuck with your pick, since your main mail program was probably the only one that could really work with its particular storage format.
(Pragmatically, only locking your system mailbox for a short period of time and only doing simple things with it tended to make things relatively reliable. Shared style mail programs had much more room for mistakes and explosions, since they had to do more complex operations, at least on mbox format mailboxes. Being easy to modify is another advantage of the maildir format, since it outsources a lot of the work to your Unix filesystem.)
This shared versus exclusive design choice turned out to have some effects when mail moved to being on separate servers and accessed via POP and then later IMAP. My impression is that 'exclusive' systems coped fairly well with POP, because the natural operation with POP is to pull all of your new email out of the server and store it locally. By contrast, shared systems coped much better with IMAP than exclusive ones did, because IMAP is inherently a shared mail environment where your mail stays on the IMAP server and you manipulate it there.
(Since IMAP is the dominant way that mail clients/user agents get at email today, my impression is that the 'exclusive' approach is basically dead at this point as a general way of doing mail clients. Almost no one wants to use an IMAP client that immediately moves all of their email into a purely local data storage of some sort; they want their email to stay on the IMAP server and be accessible from and by multiple clients and even devices.)
Most classical Unix mail clients are 'shared' style programs, things like Alpine, Mutt, and the basic Mail program. One major 'exclusive' style program, really a system, is (N)MH (also). MH is somewhat notable because in its time it was popular enough that a number of other mail programs and mail systems supported its basic storage format to some degree (for example, procmail can deliver messages to MH-format directories, although it doesn't update all of the things that MH would do in the process).
Another major source of 'exclusive' style mail handling systems is GNU Emacs. I believe that both rmail and GNUS normally pull your email from your system inbox into their own storage formats, partly so that they can take exclusive ownership and don't have to worry about locking issues with other mail clients. GNU Emacs has a number of mail reading environments (cf, also) and I'm not sure what the others do (apart from MH-E, which is a frontend on (N)MH).
(There have probably been other 'exclusive' style systems. Also, it's a pity that as far as I know, MH never grew any support for keeping its messages in maildir format directories, which are relatively close to MH's native format.)