For deterministic wallets, which key is better to give out? I am just finding out about deterministic wallets. I have read here that it is possible to derive the private key if one happens to have given out both master public key AND a child private key. 

So if we want to give someone a key to send funds to, is there a better one to give out? Or can it be any, as long as we don't give out the other one?
You give out a single address, as with a normal wallet. The master private key and master public key are both for use locally and should never be shared with any party, especially not the former.