In my book, "sending your own email" means you operate your own mail server, without using any third party relay (not Gmail's, not your ISP's).
I'm personally halfway there: I have root access to a virtual machine on the cloud (gandi.net), and configured it to relay my email for me. Next step would be to have physical control, but I kinda abandoned that idea when I learned that residential IPs are blacklisted (Hotmail even made it an explicit policy, it won't even appear in the recipient's spam folder).
Virtually no one else reads that book. To everyone else, the user operating the MUA is sending their own email.
Also, they control your right to send and receive email: if your account is terminated for some reason (they reserve the right to, blah, blah), you cannot move to a new provider, or redirect inbound mail or nothing: they own your email address.
Still think your send your own email? Fool.
Besides, there's more to it than (in)security. For instance, what happens to your various accounts tied to your email if your account gets terminated? You will not be able to redirect your incoming emails, and notifying your friends will be difficult if you relied on the web interface's address book.
Also, it's a hole ecosystem: if you use a big webmail, then you allow your provider to spy on everyone you exchange emails with. If you have the means to avoid those, it is your responsibility to do so. Surely you don't like inflicting this kind of spying on your friends, do you?
So far, the only reliable way to have proper control over our own email is to control our own domain name, and operate our own mail server. Or at least use small, trustworthy providers. In other words, "send our own email".
It's intellectual property/monopoly all over again. One will chose the terms that suits one's side of the debate. I side for privacy, security, and reliability; and argue that webmail providers do not provide any of them. This thread is the first time my "sending one's own email" wording hasn't been an unmitigated success.
The issue here is one of control: relays can basically read every mail they relay, and the likes of Gmail do. Spying on you is how they make money. No targeted ad would be possible otherwise. And the automation only makes it worse (because it scales, and can be repurposed).
And the user can do nothing about it. That total lack of control is why I maintain they do not, in fact, send their own email. Now there's PGP, but that would look conspicuous. If everyone had their own mail server, TLS alone would provide pretty good security.
What angers me the most is, even I don't have a choice: most of my friends use a big webmail provider, which invades my privacy whenever I communicate with them. This would never happen if we all had our own mail servers.
Also, it doesn't hide the authenticated user, envelope mail from or envelope rcpt to values used in the message, unless, like you said, the sender and recipient(s) are using their own SMTP servers.