Tom you said
Tom Ligon wrote:Forgive me if I'm coming in a little late here. There were large numbers of electromechanical calculators running in the 1930's
These guys used to crunch exponential series calculations all day, and the name for these poor suckers was "computers."
Tom, you are right in that electromechanical calculators and computers where the predecessors for the electronic ones for example some of the electromechanical calculators were the Autarith, created around 1902, , MADAS  and the Casio 14-A  and for the electromechanical computers the Zues Z1, Z2, Z3 and Z4  and of course the Harvard Mark I – IBM ASCC 
The use of computer to refer to a person who does mathematical calculations dates to at least 1613 
However you said
Tom Ligon wrote:Somewhere in there you have to acknowledge the Babbage Engine, an attempt at a quasi-programmable electromechanical number cruncher that probably should be classified as a programmable electronic calculator.
This is not correct because the Babbage Engines are mechanical computers  and . In addition Babbage designed the Difference engine around 1822 and the Analytical engine around 1833 to 1837 (these engines where basically never built during his life time). Electrical mechanical devices were bascially being invented from 1830's onwards
 George C. Chase, "History of Mechanical Computing Machinery". Annals of the History of Computing, Volume 2, No. 3, July 1980.
 Edward Vanhoutte, “International History of computing, A selective overview (1614-1961)”, pg 4.
 Oxford English Dictionary (2 ed.), Oxford University Press, 1989
 Difference engine
 Analytical engine
 Edward Vanhoutte, “International History of computing, A selective overview (1614-1961)”, pg 2. (Note Vanhoutee refers Babbage engines as Calculators)
 history of Electricity
Edit: Corrected citation