It is unclear what question is being asked here.

If it is a question about the history of formal accounting (within an English-speaking context), that could make the question off topic or at least difficult to answer within this community.

If it is about the semantic transition from "counting" to "record keeping", that transition doesn't appear too complicated or surprising to me; the records are essentially records of numbers that are to be added together to yield the balance of the account. If the records became a common tool to make such counting and subsequent liquidation more reliable historically, then the transition is a form of metonymy.

"If it is about the semantic transition from "counting" to "record keeping", that transition doesn't appear too complicated or surprising to me". yes, this is my question. this transition is complicated and surprising to me, that's why I asked. but thanks for assistance. can you write out your comment as a full length answer?

Let's wait whether anyone familiar with the history of accounting and banking, or with the subtlest meanings of "count" is able to answer this more confidently.

I don't feel like that Etymonline necessarily skipped many or any steps. Metonymy is an extremely common figure of speech (and many kinds of metonymy have their own special names) and many permanent new meanings of existing words have originated from a historical metonymy, rather directly. But that's a mere common principle and I'm unsure about the word "count" in its newer meanings specifically.