Help with digital keyboard

Hello,

I really want to purchase keyboard to start playing on my Pro Tools. But I
can't make up my mind on the products. First of all, I must have weighted
keys. Are weighted keys the same as hammer action? I just have to have
that real piano feel!

Second of all, I am not sure what the difference is between a synthesizer,
and a digital keyboard. Obviously I want the instrument to have many drum
tracks built in, as well as many different instrument sounds. In addition, I
must have midi so I can record in Pro Tools.

I am in love with the Korg, but I have to tell you, each time I try it out,
the weighted keys (hammer action?) feels less and less like a real keyboard.
On the other hand, when I try to much cheaper Casio the weighted keys feel
wonderful. However, the Casio does not offer all of the options (i.e.
various sounds built within) that a Korg, or Yamaha have.

Can anyone give me some pointers on what to look for in a keyboard? Heck,
the Korg does seem to be the industry standard when it comes to digital
recording, so maybe I will end up with a weighted key version. But are
there any different companies that offer the same high quality at a reduced
price?

Thanks for your help.