I have always referred to members of the council as Councilmember, Council member (depending on the use). I learned this in the late 1990s from my then bosses at DC Action for Children and peers in the field. This was the way we referred to these elected officials until recently. Recently, however, CMs and people (advocates, residents, providers) started calling them "Councilwoman" and "Councilman." To me, this is so Chicago (think Alderman) and San Francisco (Councilman when they are not being referred to as Supervisor).
Now it’s been pointed out to me that the title "Chairman" is used in the city’s home rule charter and former Council Chairman Linda Cropp made an effort to use the official, legal term throughout her tenure. Current Council Chairman Vincent Gray is also using the appropriate term but his colleagues, chairing committees, are referred to as "chairperson." Yuck.
So back to the Councilmember-Councilman issue, the home rule charter refers to members of the Council of the District of Columbia as members. This friend who did the pointing out regarding “Chairman” also shared that Ward 5 CM Harry Thomas, Jr. has taken to referring to himself as Councilman. But if you take a look at his web site, there is "Councilmember" everywhere. So this still remains a mystery to me.
So now to the other question raised, and that is about the John A. Wilson Building, home of the Council of the District of Columbia, office of the Mayor of the District of Columbia, and home to various CFO and executive branch offices.
Again, going back in time, the building has been referred to as the District Building (pre-1994) and after the renaming in 1994 as JAWB or the Wilson Building. When the council and mayor first returned to the fab building on Pennsylvania Avenue, some of us used to refer to it by the number (1350) so as to differentiate it from the "old Council Chamber" (441). Some of you may be too young or new to the city to remember that when the Wilson Building was being renovated, the City Council and EOM moved to 441 4th St NW. For a period of time after the move back to JAWB, we used building numbers to avoid confusion, particularly when we were scheduling meetings, etc. Fast forward to 2009 and you see 1350 being called "City Hall." Perhaps folks new to the work and the city whose home town had a city hall default to that. Not sure, doesn’t much matter.