I'd say you are for developing "Hybrid apps that look and feel completely native"
You might not like "hybrid" but it is, by definition, what you've built.
But I would say the main difference is OS APIs. Ability to manage memory, interact with filesystem objects etc.
mobile website = run in mobile safari, assets loaded over http
hybrid app = run in a webview, most assets stored locally, some data optionally loaded over http
native app = written in the officially supported language of the platform
What _should_ the terms mean is a very different question than what _do_ the terms mean.
As far as end users are concerned there are only two categories: native and mobile websites (though they're more likely to describe them as "apps" and "websites"). Hybrid is a term used by developers to communicate a particular technology stack, it is a subset of end-user native.