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Hydraulic conductivity (K) measured from pumping and slug tests is the Horizontal hydraulic conductivity since the inflow from the geologic formation to the well screen is  largely horizontal for both tests.

The empirically estimated K from GSA is a function of effective grain size and would represent horizontal K only for the zone/portion where the grain size was considered. To estimate vertical K you would need to consider determining K for depth discrete zones within an aquifer and establish possible anisotropy in the aquifer material. This would give an indication of what ratio Kh would be Kv (kv is often half or less than half of kh).  

It depends on the type of the test that you are conducting. Test objectives and set up are important. GSA only give the hydraulic conductivity around the measurement point. It does not distinguish horizontal from vertical HC. Therefore, it can be used as an estimated average K in all directions depending on the type of analysis one is dealing with. Some pumping tests for example in unconfined aquifers, based on the analysis and assumptions provided by Neuman, give both vertical (Kz) and horizontal (Kh) HC values. In general, if you want to get vertical HC, then vertical component of flow must be present in your test, and applicable mathematical models must be able to take it into account in their analysis. This also applies to slug tests.

Usually the vertical component of flow is ignored, but in most cases it is there, which leads to flawed estimation of flow and overestimation or underestimation of hydraulic conductivities

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From pumping and slug tests you estimate horizontal hydraulic conductivity.