What breaker configuration does my new on-demand water heater need?
I don't know what a "double pull" breaker is.  There is single and double POLE, and single and double THROW switches (that aren't actually breakers but resemble them - used in transfer switches), but not 'pull'.

The breaker marked HW is the right side of a tandem breaker and the left side is the washer (presumably washing machine not dishwasher).  It is, in short, a single POLE double THROW breaker.  (Although not really - it's actually two SPST switches on a single breaker I believe)

What this means is that there are two different lines coming off this one breaker, running two different appliances.  It is 110V and can never provide more than 110v.

IT IS NOT OK TO REPLACE THIS WITH A 60A.

In order to run a 60A line to an on demand water heater you must put in a 220V, double pole single throw breaker which can feed 220V with 2 hot lines to the new on demand heater.

So yes, you must move to a new spot - and in fact you must move to TWO spots that are adjacent, in order to get the 220v 60A you need.