In HTML5, should the main navigation be inside or outside the <header> element?

Question

It's completely up to you. You can either put them in the header or not, as long as the elements within them are internal navigation elements only (i.e. don't link to external sites such as a twitter or facebook account) then it's fine.

They tend to get placed in a header simply because that's where navigation often goes, but it's not set in stone.

You can read more about it at HTML5 Doctor.

I do not like putting the nav in the header. My reasoning is:

Logic

The header contains introductory information about the document. The nav is a menu that links to other documents. To my mind this means that the content of the nav belongs to the site rather than the document. An exception would be if the NAV held forward links.

Accessibility

I like to put menus at the end of the source code rather than the start. I use CSS to send it to the top of a computer screen or leave it at the end for text-speech browsers and small screens. This avoids the need for skip-links.

It's a little unclear whether you're asking for opinions, eg. "it's common to do xxx" or an actual rule, so I'm going to lean in the direction of rules.

The examples you cite seem based upon the examples in the spec for the nav element. Remember that the spec keeps getting tweaked and the rules are sometimes convoluted, so I'd venture many people might tend to just do what's given rather than interpret. You're showing two separate examples with different behavior, so there's only so much you can read into it. Do either of those sites also have the opposing sub/nav situation, and if so how do they handle it?

Most importantly, though, there's nothing in the spec saying either is the way to do it. One of the goals with HTML5 was to be very clear[this for comparison] about semantics, requirements, etc. so the omission is worth noting. As far as I can see, the examples are independent of each other and equally valid within their own context of layout requirements, etc.

Having the nav's source position be conditional is kind of silly(another red flag). Just pick a method and go with it.

@IanDevlin is correct. MDN's rules say the following:

"The HTML Header Element "" defines a page header — typically containing the logo and name of the site and possibly a horizontal menu..."

The word "possibly" there is key. It goes on to say that the header doesn't necessarily need to be a site header. For instance you could include a "header" on a pop-up modal or on other modular parts of the document where there is a header and it would be helpful for a user on a screen reader to know about it.

It terms of the implicit use of NAV you can use it anywhere there is grouped site navigation, although it's usually omitted from the "footer" section for mini-navs / important site links.

Really it comes down to personal / team choice. Decide what you and your team feel is more semantic and more important and the try to be consistent. For me, if the nav is inline with the logo and the main site's "h1" then it makes sense to put it in the "header" but if you have a different design choice then decide on a case by case basis.

Most importantly check out the docs and be sure if you choose to omit or include you understand why you are making that particular decision.

To expand on what @JoshuaMaddox said, in the MDN Learning Area, under the "Introduction to HTML" section, the Document and website structure sub-section says (bold/emphasis is by me):

Header

Usually a big strip across the top with a big heading and/or logo. This is where the main common information about a website usually stays from one webpage to another.

Navigation bar

Links to the site's main sections; usually represented by menu buttons, links, or tabs. Like the header, this content usually remains consistent from one webpage to another — having an inconsistent navigation on your website will just lead to confused, frustrated users. Many web designers consider the navigation bar to be part of the header rather than a individual component, but that's not a requirement; in fact some also argue that having the two separate is better for accessibility, as screen readers can read the two features better if they are separate.