We'll be in London in late July for 4 nights and in mid-to-late August
for two more nights. Because of these two times in London, it seems
like the Oyster card is probably the way for the three of us to get
around.

What does it mean to "top up your Oyster Card with a Travelcard"?
Let me see if I have this right -- We buy Oyster Cards when we arrive
at Heathrow and while making this purchase, we can then choose how
many quid we want stored on each Oyster Card and we can choose whether
we want a Travelcard also electronically-embedded on the cards as
well?

This is just how I read page 10 of:

http://www.tfl.gov.uk/assets/downloa...-of-oyster.pdf

and other such documents on the TfL website.

But it doesn't quite makes sense. How does the "touch" know whether
you want to use the Travelcard or subtract from whatever balance I
stored on the Oyster Card?

To top up a card that I've already purchased, do I take the card to a
Tube station machine (I won't have a laptop and maybe not even a cell
phone), enter whatever ID or authentication is necessary and then use
a credit card to add additional funds?

Thanks a bunch!