For POP:
That's not exactly correct. It doesn't matter which device it recieves it on first or if the website receives it first. What matters is the settings in applications and what's done at the website. As long as it's on the server, it will download to any device that hasn't already downloaded the message.
If you check at the website first and delete the message from the inbox at the website, it will remove it from the server and won't be available to other devices. IF you leave it in the inbox, it will be available to other devices for download.
On other devices, what happens depends on what setting you have stored for the server:
For Mail application, check Mail preferences under the accounts tab. Highlight the account and click on the advanced button. Inbox settings can be made here from never to imediately and inbetween.
If you set it not to remove it from the server when moved from the inbox, it will be available to other devices. I usually like to set to remove imediately when message is moved from the inbox, but that's a personal choice.
For iPhone, also check settings.
An alternative is you can also change it to IMAP for total syncing.