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I obsessed over this question longer than any one person should. I chose site/blog for better seo reasons. If you read every single article on the web on the subject as I did, you will find its about a 70/30 split. 70% saying site/blog and 30% saying blog.site
I don’t think it matters that much either. I can tell you that at CodePen, we have our blog at blog.codepen.io – the reason being that you can set different DNS records for different subdomains. This gives us the ability to host that blog at a totally different host than the root domain. So, for example, god forbid, if CodePen were to go down, the blog is still up and we can communicate through that.
In my mind, having a specific feature of a website (blog, gallery, tool, etc.) on a subdomain, creates the illusion of it beeing somewhat separate. So I would expect maybe a special layout/navigation or different colors, something like that.
Just with a /blog or /gallery it feels more integrated as a subpage so I would expect it to have like same navigation, colors as the home page.
It would depend on the way you link to it as well. (seperate link/button or integrated in the main navigation.
I hate it when you click on a “page” in the main navigation on a whole different site flips open.