Question: Running an ANCOVA is almost (but not exactly) like running the following two-step analysis: First, predict Y (the dependent variable) from X1 (the covariate), and

Running an ANCOVA is almost (but not exactly) like running the following two-step analysis:
First, predict Y (the dependent variable) from X1 (the covariate), and take the residuals from this regression; this residual is denoted by X*.
Second, run an ANOVA on X* to see if the mean on X* differs across levels of A, the group membership variable.
How is the two-step analysis described here different from what happens when you run an ANCOVA?

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