The distinction that still matters is between a gateway and a merchant account. The gateway moves the information. The merchant account holds the money after the charge is approved but before it reaches your bank. They are separate functions, historically supplied by separate companies, and now typically sold together. Knowing they are different only matters when something breaks, because it tells you which part to ask about.
Where the gateway choice affects you is in how it appears on your site, and this has real consequences beyond aesthetics. A hosted checkout sends the customer to a page the provider controls, which is the simplest option and keeps card data entirely off your systems. An embedded field looks like part of your site while the input itself belongs to the provider, which is the best of both and what most modern integrations use. A fully self hosted form, where card numbers pass through your own server, gives you complete control of the experience and pulls you into a compliance category that is expensive and unnecessary at small business scale.
The practical criteria are unglamorous. Does it support the payment methods your customers actually use, including the mobile wallets that now account for a substantial share of transactions. Does it work properly on a phone, tested by you on a real device. Does it connect to your accounting software so reconciliation is automatic. And what does it do when a card is declined, because a gateway that fails silently costs you sales you never learn about.
For nearly every first year business, the answer is to use whatever comes with your processor, choose the embedded option, and stop thinking about it.
Test the failure paths rather than only the successful purchase, since those determine what a customer experiences on a bad day. What happens when a card is declined, when a session expires partway, and when somebody presses back after paying. Each of those has a default behaviour that may be unhelpful, and a checkout that handles a decline gracefully recovers sales that a checkout showing an unexplained error does not.