Typically, the dependent variable in a linear probability model (LPM) is a 0/1-valued binary variable. What if the dependent variable $y_i$ is still binary but take on general values $a$ and $b$ rather than 0 and 1? Technically, the resulting predictor still retains its nice properties, e.g., it is the minimum mean squared error (MMSE) linear predictor. Again, we may also transform $y_i$ into a 0/1 variable; but sometimes we want to keep the original values, say, for interpretation.
My question is: Can we still call the estimator the LPM estimator? If not, what should we call it?