What is VDD in this context?
\$V_{DD}\$ is the supply voltage (i.e. the main voltage "in" to power the IC).

Quote from 1st the link you posted:


  VDD Supply voltage range of 2.375V–3.46V; VLOGIC (MPU-6050) at 1.8V±5% or VDD


As for why there's the 1.8V VLOGIC version? I suppose it's for those that want to interface with a 1.8V \$I^2C\$ bus (as opposed to, say, 3.3V SPI or \$I^2C\$ as you'd likely use with the non-1.8V version).