My road bike has a 9 speed Shimano triple with a long cage rear mech which is due for replacement. I would like to use a shortcage mech for aesthetic reasons and also for less sloppy shifts. My cassette is 13-23.
Would this present problems?

The long cage rear derailleur is needed to wrap up more chain so that you have no slack when the chain is on the smallest chainring and the smaller cassette cogs. Every RD has a wrap capacity. The capacity needed can be calculated by subtracting the number of teeth on the smallest chainring from the number of teeth on the largest chainring and adding this to the difference in number of teeth on the smallest cog subtracted from the teeth on the largest cog. In other words, if the chain is running on the smallest chainring and the smallest cog, there should be no chain slack.
You haven't offered the chainring sizes but most 9-speed Shimano triples are 52-42-30. So is that is what you have the wrap capacity needed is 32. Look for an RD with a wrap capacity of at least 32. But a short cage RD will place limits on the cassettes you can use.

I wouldn't worry about it.

Think about how you actually use a triple. I only use the granny chainring while climbing steep hills and shift out of it as soon as I crest the top. Consequently, I've never used it with any but the largest 2 or 3 rear cogs. If that's you too, do you rally need enough chain wrap for the little/little? Also, Shimano road derailleurs, whether short or long cage, are spec'd for the same size large rear cog so that's not an issue.

On the other hand, I wouldn't count on less sloppy shifts. I've used both on verious bikes and I can't tell the difference. Aesthetics, however, is in the eye of the beholder and I kind of like the way the shorter cage looks too.

Some current Shimano short-cage road rear derailers may be spec'd to wrap enough chain. https://techdocs.shimano.com/ says that the RD-5700-A and RD-4600 SS have a total capacity of 34T, and the RD-7900, RD-6700, and RD-5700 SS wrap 33T. Others may wrap 32T or 31T.

Shimano specs are also conservative. I have a RD-7400 for which the specs are 26T max cog and wrap of 28T and I have successfully run it with 50x34 and 13-30 (wrap of 33T "required"). Of course, if you exceed the specs, you need to pay careful attention to chain length and derailer adjustments.

Thanks guys very helpful answers