Curves with colliding points: logarithmic and stacky

We introduce a new notion of generalized log twisted curves, which are marked nodal curves with additional data at the marked points. In the case when the markings are distinct this notion agrees with the notion of twisted curve introduced by Abramovich and Vistoli. In addition to developing the basic notions and results, we study in this article the moduli of such curves as well as contraction maps between them. This is motivated, in part, by applications to twisted stable maps which will be studied in a subsequent article.