As a New Zealander, I wouldn’t say either of those words were similar, although I admit our usage of English is special.
My naive interpretation of those words is: perambulate is closer to just walking; rambling is a country walk with the sense of avoiding normal walking routes, but often planned and often walking with others (really need UK person to clarify).
Wander or meander-through have the sense of randomness, but lack the intentionality of deep attention to your surroundings.
Brit here: can confirm that rambling generally implies countryside walking (perhaps a bit slower and less purposefully than hiking or trekking). Perambulation sounds a bit Victorian and tends to imply walking in some sort of defined circuit. Strolling might be closer but also doesn't really imply paying much attention to surroundings.
My naive interpretation of those words is: perambulate is closer to just walking; rambling is a country walk with the sense of avoiding normal walking routes, but often planned and often walking with others (really need UK person to clarify).
Wander or meander-through have the sense of randomness, but lack the intentionality of deep attention to your surroundings.