Is monitor off setting "sticky" on today's digital camera's

I have two older digital camera's, a Nikon coolpix 950 and an
Olympus C2000. They both have some undockumented features
that you only learn after owning them for a few weeks.

One of these, the Nikon, turns on with the zoom at maximum and
the monitor on, in a way that wears the batteries down so fast that
they are hot 30 minutes later when you have to change them. In
sunlight I cannot even tell if the monitor is on or off, so I often wear
down the batteries needlessly.

The Olympus powers up with the monitor off, thereby saving
batteries. I hardly ever use the LCD viewfinder.

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So the question is, in modern cameras can you choose which state
the monitor powers up to, and if not which state does it start in.
I'm looking at compact digital cameras in the 3mpxl range to
replace one of the above that has a broken battery compartment.

Thanks