They very well may be cooking the books.  But I don't think that explains the entire difference.

Japanese social welfare has a bad reputation, but I know there are lots of city- and burrough-level services for providing jobs for the unemployed.  If you go to Japan, you will find certain jobs that seem "needless", most conspicuously, uniformed men and women at street construction sites politely waving pedestrians through carefully constructed byways so that they avoid hurting themselves near the site.  Any half-careful person could navigate these areas without any guidance, but the presence of these "waver-on's" not only clearly gives you warning that you may need to be a little extra careful around here (especially relevant for Japan's increasingly elderly population), but more importantly, they provide jobs and dignity to people who otherwise may not have such jobs -- and might be on the streets.

Japan also still has a relatively strong sense of family responsibility to elders and family in general.  People would rather take care of their own then let them wind up on the streets.  For example, my uncle drank himself into decrepit destitution through his adult life (my father is Japanese).  No doubt he would have been on the streets had another uncle's wife's resourcefulness not managed to find a way to persuade the local welfare bureau to pay him unemployment and medical insurance (even though I think he normally would not have qualified for these.)  While I imagine for any family in any developed society there would be a significant amount of shame in having a homeless family member, I believe this may be particularly so in Japan, where the principle of face is still very significant (though less over time).

Unfortunately I have no numbers to support these hypotheses, just guesses.  In short, the numbers may be underreported, but I think there are certain features of Japanese society that also reduce homelessness to a level below other developed countries'.

... all progress depends on the unreasonable mensch.
(apologies to G.B. Shaw)