You're wildly wrong.
Every single thing you just mentioned both occurs and doesn't occur with both web and native apps depending on what the app was built to do.
- Web apps can require internet
- Web apps can not require internet
- Native apps can require internet
- Native apps can not require internet
- Web apps can inform a server about what you're doing
- Web apps can not inform a server about what you're doing
- Native apps can inform a server about what you're doing
- Native apps can not inform a server about what you're doing
- You can unplug from the net, install desktop software (native app) and run it without it being able to send anything anywhere
- You can not unplug from the net, install desktop software (native app) and run it without it being able to send anything anywhere
- You can unplug from the net, install desktop software (native app) and run it without it being able to send anything anywhere until you plug back into the net
- You can unplug from the net, install desktop software (web app) and run it without it being able to send anything anywhere
- You can not unplug from the net, install desktop software (web app) and run it without it being able to send anything anywhere
- You can unplug from the net, install desktop software (web app) and run it without it being able to send anything anywhere until you plug back into the net
Source: I make offline web apps.