If your DSL connection (to your ISP) is made automatically, then the connection to your email is an additional "log on" of a sort. It would appear that what you really need is a "log off" at the email provider's site, in order to free you to log on to a different email provider(?).

You might look, on the email service window, for some kind of "log off" or "log out" button.

As a side note, any time you log on, especially at a "secure" web page, closing the browser or the browser window does not necessarily "close the connection." If you leave the site without formally logging off, the connection is still "open," and a malicious person under some conditions (i.e. with an intercepted copy of your "cookie" for the site) may be able to "piggy-back" on the connection to sneak in behind and access your transaction. A good page should time-out in fairly short order, and terminate the connection/authorization; but a lot can be done in a few moments if someone gets in where they shouldn't be.

This doesn't appear to happen often recently, but is a form of thievery not too uncommon in the past. If there's a "log off" button, it's still a good idea to use it, to make sure the connection is terminated, before you leave the site - either by going to a new page or by closing the browser.

John