What's The Difference Between The Terms Predictor And Feature

A feature may be a predictor, but it does not have to be. In $Y = f(X)$, where $Y$ is a predicted outcome, features are all available variables in $X$. So features are essentially input variables. Let us say you have a case where you have 10 features. It could be that only half of those have any predictive value. Throwing out the other features would not change the quality of predicted $Y$.

Predictor is the first level input variable, while feature may be of first level or second level. Here, first level means predictor as an input variable for predicing response or output. The second level means feature as a nonlinear transformation of the predictor or input variable using some basis or feature map to capture inherited special structure.

Feature and predictor are used interchangeably in machine learning today though I must admit that it seems that feature is being used more than predictor. The definition is the one on Wikipedia which you have already mentioned. The term predictor comes from statistics, here one definition:

An independent variable, sometimes called an experimental or predictor variable, is a variable that is being manipulated in an experiment in order to observe the effect on a dependent variable, sometimes called an outcome variable.

and my favorite definition: A predictor variable explains changes in the response.

In a nutshell: X columns: features, predictors, independent variables, experimental variables. y column(s): target, dependent variable, outcome, outcome variable.