Your observation is correct, the truth is, that anyone can open a Cyprus bank account as personal or business bank account without visiting Cyprus!
That said, you will need to make sure that you are able to provide the following 4 important documents:
1. Passport copy
2. Utility Bill not older than 3 months
3. Reference Letter from a Bank, Lawyer or certified accountant
4. a CV
If the 4 outlined points are not an issue you can open the account as personal account, for business bank accounts make sure you are able to provide the listed corporate documents here : Cyprus Bank Account - Offshore Company Formation & Offshore Bank Account - F.A.Q. | Cyprus Company Formation | Cyprus Bank Account - CCLOGIC Limited
We have sadly, seen many clients which failed to provide the required documents and therefore the process has taken several months, so it is very important to make sure you have everything by hand and that you are able to submit it at once in order to avoid any delays, decline and to visit the bank in Cyprus.... Even if you visit Cyprus, the different banks will require the first mentioned points.