3 mobile app categories exist in my understanding
This is probably more in the second category - you download the app from the app store and it exists as a real app on your phone, but it just display a webpage. It's similar to Phonegap or Cordova, but less evolved.
You just access it normally in a webbrowser, and it's coded exactly like you would code your desktop website.
No, a hybrid app is really an app - you must launch it on your phone
Most companie want to build native apps since the look and feel is much better. Some companies will settle for the second category because it's cheaper, and some companies, like a local retailer entreprise, will just make sure their website looks good on mobile (third category) since an app would be overkill.
If you want to use the same code in cordova, you should design your HTML and CSS to be responsive - making a website responsive later is longer and harder than just coding mobile-first from the start.
bootstrap, like you mentioned, is a good starting point.