There is no difference between webmail, and something that would be “real email”.
Webmail is just a way to access your email account at your email provider : you use a web browser to that effect.
The other way to access it is through another type of program, which is called an email client. Typically, Microsoft Outlook or Thunderbird.
Gmail accounts at Google, for instance, can be used as webmail, by logging to them through a browser, or through an email client. In the latter case, your email is downloaded and saved to your computer before you can read it.
Email providers need to be compatible with email clients in order to enable that. Technically, they need to provide POP or IMAP access. Most do, except some encrypted email providers, such as Tutanota, because their encryption scheme does not allow POP or IMAP.
My “one account, one email” rule is relevant to any type of account you might open on a website. An account at a web merchant, an online bank, a media site… anything. This includes an account at an email provider, of course.
You don’t necessarily need to provide a previous email address to open an account at an email provider. In fact, you should be free not to provide any.
The only good reason to provide one is to have a recovery address, which can be used for security alerts, password recovery, and the like. But it also increases your risk of being hacked.
Tutanota, for instance, which is an anonymous and encrypted email provider, won’t ask you for an alternate email address. While it’s possible you may provide one if you wish, I’m positive you don’t have to.
I don’t remember whether Google requires you to provide an alternate address. It’s quite possible, given their thirst for personal information.
If that is the case, of course you can provide a Yahoo address – and vice-versa. Any address will do. A few sites (not speaking only of email providers) will refuse addresses from a few email providers. Either because they deem them spammy, or because they are encrypted, or because they offer disposable addresses. These are mostly bad reasons, hindering privacy.
The Anon Addy service above is a different beast : it’s not an email provider, it’s a remailer. It sits between your correspondents and your main email account, and it provides you with an unlimited number of aliases, or “disposable addresses”. This is one of the best ways to implement the “one account, one address rule”.