Why is aerodynamic resistance defined inversely?
It's partly historical, partly point-of-view, but it's not a mistake.

The friction coefficient emphasises the effect of the surface on a property of the boundary layer, i.e., greater surface friction slows the near-surface wind more.  Aerodynamic resistance emphasises the effect of the boundary layer on surface-atmosphere exchange, i.e., greater mixing yields a lower resistance to evaporation.  Note that it's evaporation not advection that's being resisted.  You could turn the confusion around and argue that friction coefficient is a counter intuitive term: if there's more friction, why does evaporation increase?  This would be making the same category error.

Historically (IIRC) the electrical circuit metaphor was introduced by Penman and Schofield (1951) in an early modification of the Penman evapotranspiration equation to account for how vegetation limits the transpiration rate.  The original Penman equation is often quoted using aerodynamic conductance (or friction velocity or friction coefficient), i.e., a quantity using the the $m s^{-1}$ units that you're used to.  This is a natural choice for micrometeorologists trying to understand the bulk mass transfer properties of the turbulent boundary layer because it can be calculated from the boundary layer quantities that they measure.

Once you start introducing other surface effects on evaporation it's not really appropriate to extend the analogy of a turbulent fluid friction velocity down from the boundary layer to vegetation stomata, roots and the soil.  So we use the electrical metaphor instead, in which case it becomes slightly simpler to describe the system using resistances rather than conductances.  In the first instance it was just a case of calculating the total system resistance by adding a stomatal resistance to the aerodynamic resistance:

$E = \frac{\rho}{r_s+r_a}\left(q_{sat} - q_a\right) = \frac{\rho}{r_t}\left(q_{sat} - q_a\right)$

You can express this as conductances, but it's not as neat:

$E = \frac{g_a g_s \rho}{g_a+g_s}\left(q_{sat} - q_a\right)$

You can see in the later slides of the presentation you linked to that these resistance networks have got progressively more complex, so nowadays this neatness point might be moot.



Penman and Schofield (1951) Some physical aspects of assimilation and transpiration, Symp. Soc. Exper. Biol., 5, 115-129.