This isn't so much an answer as a clarification on terminology. Your question asks about raw, standarized, and studentized residuals. However, this is not the terminology used by most statisticians, though I note your class notes state that it is.
Raw: same as you have it.
Standardized: this is actually the raw residuals divided by the true standard deviation of the residuals. As the true standard deviation is rarely known, a standardized residual is almost never used.
Internally Studentized: because the true standard deviation of the residuals is not typically known, the estimated standard deviation is used instead. This is an interanlly studentized residual, and it is what you called standardized.
Externally Studentized: the same as the internally studentized residual, except that the estimate of the standard deviation of the residuals is calcuated from a regression leaving out the observation in question.
Pearson: the raw residual divided by the standard deviation of the response variable (the y variable) rather than of the residuals. You don't have this one listed.
"leave one out": Doesn't have a formal name, but it is the same as the class notes.
standarized "leave one out": also doesn't have a formal name, but this is not what the class notes call studentized.
Sources:
the same wiki link you have about studentized residuals ("a studentized residual is the quotient resulting from the division of a residual by an estimate of its standard deviation")
documentation for residual calculation in SAS
Best Answer
Assume a regression model $\bf{y} = \bf{X} \bf{\beta} + \bf{\epsilon}$ with design matrix $\bf{X}$ (a $\bf{1}$ column followed by your predictors), predictions $\hat{\bf{y}} = \bf{X} (\bf{X}' \bf{X})^{-1} \bf{X}' \bf{y} = \bf{H} \bf{y}$ (where $\bf{H}$ is the "hat-matrix"), and residuals $\bf{e} = \bf{y} - \hat{\bf{y}}$. The regression model assumes that the true errors $\bf{\epsilon}$ all have the same variance (homoskedasticity):
The covariance matrix of the residuals is $V(\bf{e}) = \sigma^{2} (\bf{I} - \bf{H})$. This means that the raw residuals $e_{i}$ have different variances $\sigma^{2} (1-h_{ii})$ - the diagonal of the matrix $\sigma^{2} (\bf{I} - \bf{H})$. The diagonal elements of $\bf{H}$ are the hat-values $h_{ii}$.
The truely standardized residuals with variance 1 throughout are thus $\bf{e} / (\sigma \sqrt{1 - h_{ii}})$. The problem is that the error variance $\sigma$ is unknown, and internally / externally studentized residuals $\bf{e} / (\hat{\sigma} \sqrt{1 - h_{ii}})$ result from particular choices for an estimate $\hat{\sigma}$.
Since raw residuals are expected to be heteroskedastic even if the $\epsilon$ are homoskedastic, the raw residuals are theoretically less well suited to diagnose problems with the homoskedasticity assumption than standardized or studentized residuals.