If second city status refers specifically to size and size alone then you are bang on. The 2001 census shows that Manchester is actually about the 6th biggest city, some 600k people behind Birmingham. And as you say, even when refering to the the metropolitan areas/conurbations, "Greater" Birmingham (i.e. West Midlands) is still larger than Greater Manchester ( but only by approx 200k). I cant see that much has changed since the 2001 Census to change this all that much.
However, there is an argument that suggests second city should take into account more than just population size, so it depends on criteria. Based on population, Birmingham definately is the second city, although other criteria may suggest differently
What other criteria?
People have suggested that economic productivity, peoples awareness of the city from other countries , the influence of the city on the political structure of the country, the historical importance of the city and the impact the city (and its inhabitants) have had on popular culture should all be considered, as well as a host of other criteria, rather than just the size of the place. God knows what the second city is if all this is taken into account. Im from Brum tho, so for me that will always be the second city