The 15 x 1, 705, 130 & HAx1 are all numbers for the same needle system - this is the needle used in most domestic sewing machines.
Singer's brand 15x1 sharps are labeled 2020 and their ball points are labeled 2045. (I don't know what their designation is for universals, because I don't buy Singer needles)
Thread breakage can be due to several issues, but you say you were sewing with the same needle and thread in another project, so something has changed between that project and this one - just gotta figure out what it is that's changed. Is the fabric a tighter weave on this project? Have you changed the bobbin tension? Is there something between the tension disks that's causing extra pressure or friction on the needle thread? What happens when you re-thread the machine? Anything? What happens when you change the needle for one that's one or two numbers larger?