Online payment gateway

"A payment gateway performs the same function as a point-of-sale terminal at a physical store. It allows your customers to pay by credit card on your ecommerce site. During payment processing, the gateway encrypts credit card details, validates these details, and authorizes the transaction. It ensures safe, fast payments for the customer and the merchant."

Source: Shopify, last revised on 04 May 2021

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

According to MAS (Speeches)
last revised on 04 January 2021

The [PS (Amendment) Act] will also pre-emptively address ML/TF risk that is outside of the DPT space. The [PS (Amendment) Act] will broaden the definition of cross-border money transfer service to include facilitating transfers of money between persons in different jurisdictions, where money is not accepted or received by the service provider in Singapore. That way, such service providers will come under the regulatory ambit of MAS even if the moneys do not flow through Singapore...

... the PS Act currently accords protection to consumers, whether they are payers or payees, during a domestic money transfer. As financial institutions are sophisticated entities that can protect themselves, the Act carves out the situation where a financial institution is part of the transaction. However, that means that an individual involved in a domestic money transfer transaction with a financial institution is not accorded protection under the Act. The [PS (Amendment) Act] will therefore broaden the scope of protection of the PS Act, to carve out only situations where both payer and payee are financial institutions.

Cryptocurrencies are known as "digital payment tokens" in Singapore. Businesses accepting crypto payments are considered as "digital payment token transfer service". As such, they are likely to require a payment service provider licence. 

According to MAS (Pages)

last revised on 23 September 2021

MAS regulates seven payment services under the PS Act:

1. Activity A: Account issuance service
2. Activity B: Domestic money transfer service
3. Activity C: Cross-border money transfer service
4. Activity D: Merchant acquisition service
5. Activity E: E-money issuance service
6. Activity F: Digital payment token service
7. Activity G: Money-changing service