Google Map of campsites:

Happy Campers Iceland Campsites - Google My Maps

The 2015 rules mostly affect vehicles such as camper vans.

in general you can pitch a conventional tent

When do I need permission from the owner of the land?

Land owners of private land can restrict access to and prevent camping on their land. If a land owner prepares a campsite on their land they can direct people there and is fully in his right to charge people for pitching tents there as if it was a normal campsite.

| Reykjavík Campsite

Prices for camping at Reykjavik Campsite | 2020 pricelist