The only difference between the Eternal Collection and the Battle Chest is the Eternal Collection includes the Necromancer Pack. See my thread here:
You can also download the Starter Edition for free to see if you like the game first.
If you purchase Diablo III and decide to have more content, you will only pay to add Reaper of Souls. This will leave you with the option to add the Necromancer Pack later, if you choose and only pay for that. That’s because your Account will show you already own Diablo III and Reaper of Souls. You won’t have to buy anything a second time.