On the ST the standard boot was Low (320x200x16) when using colour monitor (TV). If you are using a Mono monitor then the ST booted in High (as this is the only res it can use).
On the ST the Desktop saves the RES but this only came into effect when the Desktop booted, i.e. the loading of utils from Auto folder would always be in Low (or High as above).
I am quite sure that the Falcon and the TT (unless a TT high res monitor was on there) both did the same by default to retain compatibility.
I am quite sure the TT has no way of changing this, but if people say it can be changed on the Falcon then I believe them.