In the context in which you are using the condition number, neither number is interpretable in itself, but is always to be compared to a threshold $t$. So long as one uses the threshold consistently, it doesn't much matter which one of the criterion is used.
@whuber comments that in many contexts, the actual value, too, can be interpreted. One interpretation that is related to the the computation of covariance matrices (and their inverses) is from computer science.
The following interpretation of the condition number is often used as a rule of thumb in numerical programing. If you define the condition number as:
where $\lambda_p$ ($\lambda_1$) is the smallest (largest) eigenvalue of the original matrix and $\log$ is the base 10 logarithm, then $k$ can be interpreted as the number of digits of accuracy one may lose on top of what would be lost to the numerical method due to loss of precision from arithmetic methods.