Agree.

Подниматься is a more abstract word, it means "ascend".
Лазить is a more specific word, for situations like climbing a ladder, but not limited to, this word is used for many "awkward" ways of moving, for example one can say "лазить через окно" = "get in through the window", or "лазить по чердаку" ("hang out in the attic?"), or, as was said "лазить по деревьям" (= climbing trees)

Climbing up a ladder is an "awkward" (i don't know how to describe it) way of moving, therefore we use лазить.
Walking up a staircase is a simple walk on foot, there fore the more general подниматься. However if someone is drunk, then лазить can be used: он был так пьян, что еле лез по лестнице = he was so drunk that barely was able to "climb up" the staircase.