What international payment systems do exist? I recently read up on the Society for Worldwide Interbank Financial Telecommunication (SWIFT). As faar as I understood it was up until recently the only provider of a system to exchange money internationaly between different entities.

(There are other national systems)

Now I saw that china is developing its own system called Cross-Border Inter-Bank Payments System (CIPS). This system is still using SWIFT to communicate with certain entities but can also connect them directly.

Also the Instrument in Support of Trade Exchanges (INSTEX) is beeing created by France, Germany and the UK. This system is not connected to SWIFT and currently only allows goods to be traded.

Did I understand it right that those systems are partwise competitors and are there more such systems?
SWIFT, as a private company, only provide secure transmission of financial messages between the participants. The message types or so vast from customer payments and cheques to financial institution transfers, treasury markets, commodity transactions, Letter of Credits etc. Providing instant, secure, without any disconnection messages together with standardization of these messages are a big job for smooth payments. 
SWIFT does not hold your money, my money nor does it keep an account for the beneficiary bank or remitting bank.
The main underlying money exchange is done through correspondent banking accounts between financial institutions. These accounts are called as nostro and vostro accounts. The debit or credit from and to these account will be confirmed through SWIFT messages. Before SWIFT, the Telex system is used for communication. 

Chinese, CIPS is used only for Renbinmi transactions and is not a competitor to SWIFT. The member banks to this system can use it as a small network but there may be still many banks who are not a member of the system.