The term "bias" has a specific definition in the statistical literature (the difference between the expected value of an estimator and the thing being estimated), but that isn't to say it loses its original, more general meaning. Which one is intended will depend on context, and oftentimes you will have a mixture of the two.
I would say the first usage is in general the less precise kind, as data imputation is a method that's used in applied problems where one need not assume that any true value of the parameter even exists. Here it's basically synonymous with "shrunk towards zero."
As far as the second usage is concerned the term bias-variance trade-off does originally derive from the more formal definition of bias, but nonetheless I would still say this refers more to the general "inflexibility" of a model fitting procedure, and not necessarily the question of whether or not an estimated regression function is correct on average.