(Very superficial answer, I haven't looked deeply, please feel free to correct me.)
Перенести derives from нести — to carry. The primary meaning is that you physically carry something from one place to another. But it can also be used (idiomatically?) at least in the following cases:
- carry (transfer) data/information from one location to another (e.g. “я переніс дані зі старого комп'ютера на новий” — I moved data from old PC to new PC).
Перевести derives from вести — to lead. The primary meaning is that somebody follows you and in that way you make them to change their location. But it can also be used (idiomatically?) at least in the following cases:
- make something to change its state, turn something to other mode, ascribe somebody to other category (e.g. “перевести війська в режим бойової готовності” — to put army into combat readiness).
Гриценко перевів у надлишкове майно 919 танків…
It's not about moving tanks, it's about reclassifying as a “excessive property”.
Як перевести гроші з Польщі в Україну.
For wire transfers:
перенести гроші (the worst idea):
- You could say “як перенести гроші через польсько-український кордон” (how to carry money across the Polish-Ukrainian border) — but that would mean literally carrying bags of cash (not a wire transfer) on your feet (not even using any transport; (пере)нести in its original meaning relates only to carrying by feet; for carrying via any transport use (пере)везти).
- You may probably try extending data/information transferring semantics to virtual money in some cases (something like “я переніс усі свої заощадження в надійніший банк” — I moved all my savings to more reliable bank), but it's a quite limited set of cases (particularly, it must be obvious from the context that we are talking about non-cash money, because otherwise the physically-carrying-cash-by-feet connotation will prevail). In the discussed quote it would sound just weird.
перевести гроші (probably not the best but sometimes used way):
The state-changing/mode-turning/category-ascribing semantics can probably be extended to virtual money — i.e. like “a wire transfer is ascribing the money to another account”. Перевести гроші is sometimes used and even is mentioned in some old dictionaries; however modern style guides often discourage its usage (1, 2, 3). (Maybe перевести гроші can be considered as an obsolete way to say about money transfers that is still alive only due to Russian influence, but I can't say that for sure, maybe it's still perfectly correct but just not so often used.)
Казати means to say, to tell, and переказати means to retell, so it's not fully clear for me how the meaning of переказати гроші arouse. Never-the-less переказати гроші is the most popular way to say about wire transfers nowadays.
Як перевести бізнес в онлайн?
That's about transforming your business (so, again, category/mode/state changing).
Як перевести оплати в телефон і почати користуватися онлайн-банкінгом…
Probably somewhat poorly-worded way to say that you change the way you make your payments, i.e. transform your payment-related habits.