barotropic component definition
The top expression is basically the depth-averaged velocity. Meanwhile the second expression only applies to geostrophic flows. The depth-averaged velocity can be caused by many mechanisms, with the geostrophic balance being one of them. The most common ones are the wind and the tides (but also differences in atmospheric pressure, sloping bottoms, ...). Both tides and wind create a barotropic pressure gradient that has to be compensated by other forces in the Navier-Stokes equations, with friction being the most common one. The geostrophic balance is a specific solution of the NS equations that neglects the effects of friction and any other forces. John Wilkin (Rutgers) has some good notes on how to estimate these currents from observations.

The barotropic pressure gradient is also the integral result for the entire water column. A good illustration is the thermal wind example:  You need a difference in slope between isobaric (pressure) and isopycnal (density) surfaces to give you a baroclinic gradient. In theory in a barotropic fluid the flow is the same at all depths: $u(z)=u_{bt}$. Clearly, in practice, any fluid will have boundary layers in which that is not the case.