Is there a standard definition of a "dry-down" event in hydrology?
Dry down is mainly dependent on the speed where a local water storage would be depleted, considering those components extracted from the simplified water balance equation


water input (precipitation)
local drainage capacity (runoff, infiltration, percolation)
evapotranspiration
storage change and capacity (pores in the rock and soil)


To generalize your question 


  is it relevant how much rain happened before the event?


I would say yes, as all input / output and potential changes in the storage need to be taken into account to calculate the balance, and ultimately the moment of depletion of the reservoir and thus the beginning of the dry-down.

Antecedent conditions are of an utmost importance when estimating a dry down moment.If the storage was relatively empty prior to a rain event, it could take a lot of input to recharge it, this without neglecting the other factors. 

To translate as an example, a huge rain event when the reservoir was empty prior to the event could recharge the reservoir and provide some surface runoff. A similar rain event when the reservoir was half-full prior would recharge for a while and then flood the area because of the important water excess.

This is dependent on the climatic zone (ex Köppen classification), say a warm and dry desert region will have a quite different balance than a polar region, because various input sources and different underground geophysical properties. Yet, within each regions, much variability will be encountered locally, depending on


the ground/soil porosity (sands or weathered rocks can store more water. A clay layer will prevent downward infiltration)
the presence/absence of vegetation (vegetation facilitate percolation/infiltration)
the temperature of the soil : permafrost hydrology is quite complex - generally cold permafrost have a poor drainage capability
maybe a floodplain will contribute to ground recharge one year, but maybe not the next, depending on the current year precipitations and antecedent conditions
plus many other factors etc...


It may be possible at the scale of a single watershed to do a prediction model on the probable schedule of a dry down - or a restricted land unit, because a lot of drilling and geophysical all around surveys would be necessary to know the ground properties, the storage properties. Also a couple of monitoring station (discharge) at higher stream order segments would be required. Rain meters stations too. Moisture would be monitored in representative ground units.

But this would apply locally, and be hardly exportable at a larger scale, in my opinion. I would go with this statement of yours:


  Or is it all arbitrary and defined differently for each individual piece of research?


yes generally