It simply means that I aim to learn all parts of our stack and do my best to master them. It's not a chore to me. I don't have to kick myself to keep up with design trends and programming trends. I love both of those fields and can't even imagine not keeping up with both side of the coin.
In my team there are some very senior back-end programmers. There's also some very senior front-end developers and very senior designers. Do I consider myself above them? Like some sort of unicorn rockstar? Nope. I'm simply a good support player.
I believe that a good full-stack developer is simply someone who is able to work on all parts of a project and help facilitate the communication between the different fields. This last part is the most important of all.