My wife has always believed that if you touch both (and maybe just one)
terminal of a flat disc "button" battery (eg CR2032, CR1620 as used in a
watch, key fob, PC BIOS), you will cause the battery to discharge, so you
must take care when fitting a new battery not to "short the terminals with
your fingers".

I can imagine that sweat on the metal terminals *may* cause a bit of
oxidation over a long period of time, but I can't see how a body resistance
of many kilohms will cause the battery to go flat.

Is there any truth in what she's always believed and what she advises me to
do?