Since reading in one of your earlier posts your ideas about this being connected with an escort request (rather than a celebration of having laid an egg), I have been watching my flock with it in mind, and my experience is consistent with your hypothesis. My hens call for an escort whenever they've laid in the coop and the rest of the flock is not in the vicinity, and the rooster (I have only 1) answers promptly and comes running, then the pair return whence he came, with or without him trying to mate her (he has no competition). If the flock is in the vicinity, the hen emerging from the coop after laying does not 'sing' but just joins the rest quietly.
Unfortunately many aren't really applicable to the backyard chicken keeper with regard to how they keep their chickens. The nest box article may be, but others are fairly specific to free range flocks.
Thanks for reading the article.