Bias and Variance definitions
I would say that both the definitions you gave above refer to variance. 

Variance is an index of model complexity. Thus:

high variance -> high complexity -> many degrees of freedom in model -> it overfits training data -> if dataset changes, accuracy degrades a lot

Bias is an index of the systematic error of the model, as it shows the difference between the expected value of various model's predictions for a given target output and the ground truth.