To be honest I am not sure how it works when you use your data connection (3G or 4G) to then connect to the internet.
But you are right in saying that if you use your 3G or 4G rather than wi fi then it IS using your data allowance.
I may have got the wrong end of the stick here, but it sounds to me that you are trying to use the internet in a way that it cant be traced back to you (sorry if I am mistaken).
If so then be aware that a combination of your phone company and whatever ISP you use will have a track of everything you do on your phone and on the internet so nothing is truly private.
If you want to use the internet at home without being traced I believe you can use something called a VPN (Virtual Private Network).
However I have never setup or used a VPN so will leave it to others to explain if that is what you need and how they work.
An intro to VPN here (there are other web sites that explain it all )
https://www.bestvpn.com/vpns-beginners-need-know/