I have to admit I've tended to have the DWP/HMRC definition of a 'single parent' in mind when I think about the term, as I've had to be so careful to not overstep the mark into 'living together as husband and wife' to maintain my tax credits! In their terms, LTAHAW doesn't actually mean living sleeping every night under the same roof, it can mean spending some nights together and also sharing meals, shopping etc. They don't take into account whether your partner (who might not be your child's father) takes on any parenting duties with your child!
I see myself as having been a single parent for 10 years. I got a lot of help with parenting from my parents and sisters, but DS never had contact or any maintenance from his dad. I don't think you have to be coping with everything completely on your own to be regarded as a single parent, I don't like the idea that the fact that I'm fortunate to have had the support means that I'm 'less' of a single parent than someone without it. I know other parents who have had lots more support than I have, probably more than many mums who are married (did well out of divorce, big family/friends network around to help) but they are all still definitely single mums.
I think I link the 'single' bit to the relationship rather than parenting - because that's what the DWP do, I suppose. I do find it acceptable for non-resident fathers to call themselves single dads, I don't find that odd at all. I normally expect them to be non-resident, as most of the single dads I've met have been.
I have a DP now but I still regarded myself as a single parent until he moved in. He never had sole care of DS before then, although we'd go for days out and spend lots of time together, and he'd listen to my concerns/moans. But we didn't become a unit until we lived together, and started sharing finances, chores, responsibilities etc. I don't see myself as a single parent any more (the DWP would have words with me if I ticked that box!), although for the major decisions like schools, DP isn't involved.