Nature of payment gateway

They are possibly regulated by the Payment Services Act as providing a merchant acquisition service and/or a domestic money transfer service. If there is fund transfer to or from out of Singapore involved, cross-border money transfer service may also be involved.

According to MAS (FAQs)
last revised on 31 March 2021
Additionally, note that

14.1 [A]n entity provides merchant acquisition services where the entity enters into a contract with a merchant to accept and process payment transactions, which results in a transfer of money to the merchant. Usually the service includes providing a point of sale terminal or online payment gateway. 

14.2 A payment service provider that operates a closed-loop e-money account issuance service, where transactions can only take place between users of the particular e-money accounts issued by the payment service provider, may also be providing a merchant acquisition service if one of the users is a merchant and there is the requisite contractual relationship mentioned above.

Domestic money transfer service are "services that provide fund transfer services in Singapore. This could include payment kiosk or payment gateway services."
Cross-border money transfer service are "services that provide for inbound and outbound fund transfer remittance services in Singapore."
Source: Singapore Legal Advice, last revised on 21 September 2021

Note that all providers of payment services must obtain a licence to provide any of the services covered by the Payment Services Act, including merchant acquisition.
Source: MAS (Pages), last revised 30 June 2020