I think that the terms (forest, mountain, open air, deserted house, ...) already describe inner states rather than external places. Bhikkhu Buddhadasa names in "Heartwood of the Bodhi Tree, page 67) "forest" (arannasanna) as a state preceding the arupa jhanas.
Some of these secluded spots are also found in non-Buddhist traditions - think of "Moses on mount Sinai", the "sermon on the mount", in the Bible for example. Or the "root of a tree" in Shamanic traditions. They describe inner places (states) of seclusion and as such are the ground on which the mindfulness meditation can be done. We have to remember that there was no Buddhism at the time of the Buddha, so he had to build his teachings on the ground of meditation practice of all the other traditions around.
So I think the actual suttic instruction is: bhikkhu, enter one of the secluded states you know from your previous meditation practice of whatever tradition you have been taught in and THEN start with satipatthana as the Buddha teaches.
And seriously, that is all the instruction one needs.