Which is correct? If both are correct which one should I learn to use?
I am Polish native, and I checked the bing, google and Ivona pronunciations. Yes - they sound similar. Personally I would choose between them the Ivona with Jacek voice, but not much difference - maybe "Piję" he speaks a bit more clear. All of them pronounce "kawę" like you said. It's common usage in everyday, "fast" Polish. Howewer it seems that your Michel Thomas Polish shows example of nicer and more elegant pronunciation of this word. But still - both should be consider as correct - select which will be more comfortable for you to speak.