Do toddlers "get" Christmas?

Recently, several people have mentioned to me that Greta must be so excited for Christmas this year, now that she actually "gets it." 

It has made me wonder whether my child is more oblivious than most, whether others' understanding of a 21 month old is inaccurate, or whether there is more that I should be doing to "prep" her for the magic of the season. It is most likely one of those things, though, because my child definitely does not "get" Christmas. 

The closest we've come to that is that she has gazed in wonder at the Christmas tree a few times and has furtively touched the strand of beads that wraps around it. I guess she's also pointed at the nativity scene on our mantle and vocalized some of the animals' noises. She also is obsessed with a board book that my mom gave her that is basically the song of Jingle Bells set to pictures of adorable mice riding in a sleigh. It is fun to sing to her and certainly makes me feel like I'm involving her in the festivities more, but she just doesn't understand that there is anything particularly remarkable that she should anticipate on Christmas morning, much less, the Christmas story or anything of that ilk. 

Should we be doing something differently? 

She definitely got into the spirit of things on Halloween. We dressed her up as a puppy, which she mostly ignored, but she very much enjoyed the ritual of waiting for the children to arrive at her grandparents' house, going to get the silver bowl of candy, etc. I don't think she would have "gotten" Halloween without the tutelage of her grandmother, though, who explained the procedure to her while they waited at the door. 

So should we be doing the same with Christmas? Telling her the Christmas story? Showing her the stockings and explaining that treats would appear (we have not yet bought any presents for her stocking -- oops) on Christmas morning? Or is she too young to understand this in advance and it is best to explain it in-process, the way that she was explained the rituals of Halloween? Or, should we appreciate the fact that our child currently has no expectations of this holiday -- whereas she likely will for the rest of her life -- and enjoy the fact that we don't have to stress about meeting her expectations?