Oyster vs Travelcard - Family of 4/ Four Days in London?

Zone 1-2 Oyster Card/ Buy one way tickets from Heathrow

No such thing :-) The Oyster Card is a preloaded debit card which gives you some discount compared to cash - it is not linked to a specific set of zones. Buy your Oyster Card in Heathrow and use it to go to London.

am I able to add any amount to an Oyster card

Yes - but if you put in too much you can get it back before you leave London.

Thank you so much!

So, even for the 12 year old, the Oyster is better than the travel card as long as I have someone add the Young Person discount, correct?

gefam, You appear to have made the standard misinterpretation of what an Oystercard is.
"Zone 1-2 Oyster" - An Oyster Card is merely an electronic wallet, a container of money.
Into the Oyster Card you can either put credit, which is then used to pay for your trips, this is called "pay-as-you-go".
Or, you can put a daily, weekly or longer ticket for one or more zones.

Watch this video: https://www.youtube.com/watch?v=LlZ_xDx2Zl0

For most tourists, the "pay-as-you-go" option is the best. Using an Oystercard the price is much reduced from buying individual tickets, PLUS you don't have to buy tickets (and you can't buy tickets for buses), just touch your card and get on.
PLUS there is a daily "cap" once you reach a certain amount of trips, everything after that is free.
Info here: https://tfl.gov.uk/fares/how-to-pay-and-where-to-buy-tickets-and-oyster/pay-as-you-go
Here: https://tfl.gov.uk/fares/
and here: https://tfl.gov.uk/fares/find-fares/tube-and-rail-fares/pay-as-you-go-caps

So, when you arrive at Heathrow, buy Oyster Cards for each person, and put some credit on each card (£20-£30).
And use these cards for all your trips (tube, trasin, bus). When (if) they get empty, top them up.

Get the pay-as-you-go Oysters at the airport Tube station. They have manned windows there and will be able to set up the children discounts for you. Not sure how the "free" one works, never had to do that. Yes, buy regular Oysters, Visitor Oysters do have some limitations (no child discounts was one) and no real advantage. For the length of time you are planning, a pay-as-you-go Oyster would be your best option, in my experience, and Travel cards (either paper or loaded to the Oyster) probably will not save you any more.

You can add any amount to an Oyster at any time at any Tube station at the ticket machines. I usually add my coins to it when I feel I have too many in my pocket and the Oyster is getting low. No need to keep a huge amount on the cards and if you have £10 or less on them when you leave London you can cash it in at any Tube station if you don't want to keep the cards for your next visit.

I always ride the Tube from Heathrow. It is only about three pounds on the Oyster at off peak times, much less than a separate ticket and very much less than the Heathrow Express train. Takes about an hour depending on your destination and provides some very interesting people watching.

Using an Oyster also removes any worry about which zone you are in. You will never be charged more than the daily cap price for the zones you go through on any day and the Oyster knows which zones you have been in automatically. This is much simpler than trying to determine in advance which zones you might want to get a Travel Card for and miss because then you will have to pay the overage before you can exit the Tube station.

An aside about Oyster:
Use the card by tapping it at the entrance gate of your station. Then tap again to get out of your destination station. The correct fee will be deducted, and it sure is faster than fumbling with cash or the old tickets.

In the links above, you can read about the daily limit, after which any rides are free for the rest of the day. Buy your Oysters from a clerk at Heathrow; the personnel can be help you choose

Got It! Thanks to everyone for all of the tips and clarification. Every country/city is a little different so I am trying to get all of my info straight ahead of time! You all are the best! =)

And based on something that seems to confuse tourists...

On the tube and other trains you touch your Oyster to the gate sensor when you enter or leave. However on buses only touch your Oyster when you get on, if you touch when you get off you will get charged twice.

You may have seen other posters refer to loading the Z1-2 travel card onto an Oyster card because if you are staying for the 7 days and using it intensively there is a saving over the daily rate you get with PAYG.

With as few as 4 days this isn't going to pay off.

You may be able to use contactless payment, if you have it, for the adults but you can't get the Young Person discount that way.

Maybe worth mentioning in passing that your 10 year-old only gets free travel if accompanied by an adult.