This Google search turns up an interesting result.
What follows is my speculation.
A deterministic model is a "Mathematical model in which outcomes are precisely determined through known relationships among states and events, without any room for random variation." From: What is a deterministic model?
I think the coefficient of determination can be thought of in terms of what is determined/known (without variation) about a model. If $R^2 = 1.00$, then 100% of the relation between the predictors and the outcome is determined. If you put in values for your predictors, you always get a prediction with no error term. You know the outcome with no room for error, or variation of any type. Your model becomes akin to $2 + 3 = 5$.
Typically, you do not have 100% $R^2$, so even though we obtain the same predicted value for two cases with the same values on our predictors, there is some error. So the correct predictions for both cases remain unknown and the correct predictions are probably different. But these correct predictions are determined (or we know them) to some extent - the extent of the $R^2$.