There is no problem with using the word "predict".
It is important to recognize that predictions are unrelated to causality. Consider a case where most people who die in a hospital emergency room die of a heart attack. If you hear that a patient died, but didn't know the cause, you could predict that it was probably from a heart attack, because you know that heart attacks are responsible for >50%. You are making a prediction, but you are predicting an unknown cause from a known effect. Also, the prediction in this example is categorical, so it is analogous to logistic regression. (The analogy is probably stronger to multinomial logistic regression, but that doesn't matter here.)
For what it's worth, predictions don't have to be related to any direct causal connection at all. You can make a prediction based on a spurious correlation, so long as the relationship is reliable. Consider predicting the unknown height of an identical twin based on the twin's sibling. In this case, both heights are effects of a set of common causes (shared genetics and environment). The height of neither twin is a cause or an effect of the other. Nonetheless, you can make very good predictions in this situation.