Like @Scortchi and @whuber, I prefer shorter excerpts. (Besides aesthetic preference, I suspect they are more likely to be read.) I think excerpts should mostly proffer tag usage guidance, and that more substantive and detailed information is better placed in the full wiki.
That said, what is "shorter" exactly? To some degree "shortness" is in the eye of the beholder. Moreover, it may not be fully possible for a reader to infer how a tag should be used without a brief note clarifying the nature of the thing in question. As a result, I don't object to including some of that when: (a) appropriate, (b) made as briefly as possible, and (c) placed after the definition / basic usage. That is, it would be somewhat less prominent, in what newspaper reporters used to call the inverted pyramid style.
To put this more concretely, when the question is asked in the abstract, I would have responded as @Scortchi (+1). Indeed, I have edited excerpts to shorten them and move information into the full wiki. On the other hand, in the example given in @amoeba's answer (
[elastic-net]), I think the excerpt was perfectly appropriate and not excessively long.
As I say, I prefer shorter excerpts, but sometimes some brief information about the topic is necessary to help people could understand how the tag should be used. For example, I wouldn't find "a regularization method for regression models" helpful for the elastic net, because that is equally descriptive of the LASSO and ridge regression. Mentioning orthogonal and Demming regression in TLS could help people recognize that it is the tag to use. Likewise, "a measure of statistical dependence..." is too nonspecific for my taste.
My suggested text for the three excerpts in question is below. I do believe that every character you can trim increases the likelihood the excerpt will be read. That said, I have attempted to retain as much of the informational content as possible. I will acknowledge that these may need some work before they are posted; I'm not familiar with distance covariance and only somewhat with total least squares.
A regularization method for regression models that shrinks coefficients towards zero. The elastic net combines the penalty terms used for LASSO and ridge regression.
The numbers of characters are 326 (amoeba) > 165 (gung) > 46 (whuber).
A method to fit a linear model by minimizing the errors in both X & Y (OLS minimizes errors in Y only). Orthogonal and Deming regression are special cases of weighted TLS. It is often used when both Y and X have measurement error.
The numbers of characters are 425 (amoeba) > 230 (gung) > 113 (whuber).
Distance (aka Brownian) covariance is a measure of statistical dependence between two random vectors of any dimension. DC(X, Y) = 0 iif they are independent. Sample DC is computed via the pairwise distances between all points.
The numbers of characters are 431 (amoeba) > 226 (gung) > 106 (whuber).
Update by @amoeba
Following some additional discussion in the comments in this thread, we converged on the following excerpts:
A regularization method for regression models that combines the penalties of lasso and of ridge regression.
A technique to estimate parameters $\beta$ of the linear model $Y=X\beta$ when both $Y$ and $X$ are subject to measurement error. Includes Orthogonal and Deming regression as special cases.
A measure of dependence between two random variables (or two random vectors of any dimension). Also called Brownian covariance.
109 / 174 / 127 chars. I am accepting this answer as this level of detail roughly seems to be the consensus.