In the Haggadah, in Dayeinu we say the final lines about Hashem's building for us (itself a curious idea) the temple. But different texts have different wording.
I know that in musaf davening for the 3 regalim (using my Artscroll), the phrase "la'asot chovoteinu b'veit bechiratecha" appears and Yerushalayim in general is called "beit mikdashecha" (the home of your mikdash) whereas in musaf for Rosh Chodesh, the phrase is "uva'avodat beit hamikdash nismach" (and then later the same reference to Jerusalem).
Is there a functional difference between referring to a beit habechira and a beit hamikdash that would explain any particular textual choice? Did one develop as a recognized name earlier or are they pointing to different specific things?