You probably have already read this page: https://bossarea.com/boss-dm-2-delay/
Or this one with a little bit more info about the DM-2: https://www.henrikhansson.com/product/dm-2-delay/
And according to the Boss Book ( witch I don't have, but can be "read" in this video: https://www.youtube.com/watch?v=26Fo2mStYW0 ), your DM-2 is from the 2nd last month of production, as you say JAN 1984 = 38XX00.
The compander IC in your pedal shows 8343 as the code of the manufacturing date, and that can be read as 43rd week of 1983.
Also it's PCB indicates that it is a B (3rd) generation, so it's supposedly to have a green label (must say I never saw one), and always according to the Boss Book.
But, the Boss Book have a lot of inconsistencies, just yesterday I discovered one more, my PN-2 have a serial ZB20107 so it is from MAR 1990, and the book says that the first month of production was MAY 1990.
This is not the only pedal I have that shows a date of manufacturing that is previous then what the book says it should be.
Since I look for the earliest units, even trying to acquire the first batch ones, I have a few like that, and some are from a month or two before to what the book indicates.
So, who knows?
Just enjoy it.