Here's one way that you might regard a median as a "general sort of mean" -- first, carefully define your ordinary arithmetic mean in terms of order statistics:
$$\bar{x} = \sum_i w_i x_{(i)},\qquad w_i=\frac{_1}{^n}\,.$$
Then by replacing that ordinary average of order statistics with some other weight function, we get a notion of "generalized mean" that accounts for order.
In that case, a host of potential measures of center become "generalized sorts of means". In the case of the median, for odd $n$, $w_{(n+1)/2}=1$ and all others are 0, and for even $n$, $w_{\frac{n}{2}}=w_{\frac{n}{2}+1}=\frac{1}{2}$.
Similarly, if we look at M-estimation, location estimates might also be thought of as a generalization of the arithmetic mean (where for the mean, $\rho$ is quadratic, $\psi$ is linear, or the weight-function is flat), and the median falls also into this class of generalizations. This is a somewhat different generalization than the previous one.
There are a variety of other ways we might extend the notion of 'mean' that could include median.
Both conventions you mention are ambiguous - for example you can't tell whether the number after the $\pm$ is a standard deviation, a standard error, or the half-width of an interval. (a problem that has led to numerous questions on this site).
If you're publishing somewhere that offers precise guidelines follow those, but otherwise I suggest being explicit, for example: $``\text{median } 28.5,$ $\text{ mean } 31.2,$ $\text{ standard deviation } 3.6\!"$ -- then there's no possibility of misinterpretation.
Best Answer
Rather than compare means/medians why not fit a model to each and compare the distributions of parameter estimates for each group? This approach would seem to provide much more information.
Edit: I moved my "answer" to this question as it seemed somewhat offtopic here. How to fit this neuron firing model with R?