Chasis reset button vs remote power recycle

Solution 1:

The reset-button is much preferred as it is electrically speaking much easier on the components in the server.

The reset-button on the server will usually just reset the motherboard. So the disks keep spinning (no power-cycling there) and the motherboard will reboot.

A smart-relay (or the on/off button on the server) will cut power completely which will also cause the disks (and the power-supply) to power-cycle.

As others already stated: Data-loss will most likely not occur in either case as disks these days are constructed such that a head-crash doesn't happen anymore.

Off course: If you application is hosed to the point of requiring a reboot it is anybody's guess if the data used/generated by that application can be trusted to be consistent. I would worry more about that, than about any potential damage due to the reboot.