That's the interpretation that would apply if you were trading in e.g. equities. The stock exchange doesn't come to you; you come to it. (And therefore, you need to have a legal presence in the country where the exchange is located, with bank+investment accounts registered there, holding deposits denominated in that country's currency. You can't buy stocks from the NYSE using a European bank account, let alone one holding Euros — you have to first fund an American bank account with USD; then transfer that into a special American [government-]registered investment account; and then use that account to buy the stocks.)

In this scenario, the foreign investor is in a sense creating a legal "proxy person" — a virtual citizen of the country where the exchange exists. The foreign investor is then telling the proxy person to do the trades for them. If the pretend proxy person breaks the law, the real investor gets in trouble... but in ways that are more similar to the ways corporations get in trouble, than the ways citizens of the country get in trouble.

(And some countries don't even allow the creation of these pretend proxy persons; instead requiring you to employ, and hand your money over to, a real proxy person, who will face the domestic legal consequences for executing your trades! This is how e.g. Bahamian shell corporations work; it's also how foreign real-estate investments work in the Philippines. It's never a good time.)

"The purchaser representing themselves legally in the country they're buying from" is also how the aforementioned import arrangements work. The importer registers an export company in the source country — essentially a proxy-person, though in the shape of a corporation — and the export company is what buys the goods and sends them to the import company.

Most trade isn't like this, though, because most trade isn't between legal peers (like an exchange and an investor — both basically "typed as" corporations for the sake of the trade, even if one is a sole proprietor); but instead is between one large company and a potentially-huge number of individually-legally-naive individual consumers.

It is impractical to ask all the people who want to buy a product, to create a foreign trading presence for themselves in the country of origin of that product, as if they were a professional importer. Governments don't want to maintain — or especially to validate and audit — huge databases of hundreds of millions of foreign individuals trading locally. Banks don't want to have to open and maintain hundreds of millions of tiny, individual, unprofitable(!) foreign-deposit accounts for those individuals to use to operate legally in the country.

So instead, governments and banks choose the far simpler route: making the foreign corporations register themselves in the markets they operate in. There are far fewer corporations to keep on the books for both the governments and the banks; and it's far easier to manage their accounts (because corporations are on average more competent at that kind of thing; because unlike individuals, corporations can afford to pay ~hundreds-of-dollar levies to maintain the systems tracking them and keep the bank-accounts revenue-positive for the banks; and because corporations will coalesce bank transfers, turning billions of little daily transactions to individuals' foreign-deposit accounts, into just a few daily transactions to each corporate foreign-deposit account.)

Ignoring the pragmatics, though, there's also a key distinction even in theory: when a company X from country A, trades with consumers in country B, company X is seeking to pull money out of (or put money into) a credit card or bank account that exists in country B (and which is denominated in the currency of country B), and move that money to/from country A, where it will then get exchanged for the currency of country A and drop into the company's country-A bank account.

The manipulation of the country-B bank account, can only legally be done (for most values of country-B) by a company legally registered in country-B to operate on the payment networks of country-B. There has to be a payment server sitting there physically in country-B, connected by dedicated line to the inter-bank network the banks of country-B use; and that payment server has to have signing certs installed to emit messages on the line that will be accepted by the network — signing certs which required a bunch of legal and contractual and trust-handshake hoops to be jumped with the banks of country-B before they'd be issued.

So these are the only options for getting access to such a payment server. Either:

- company X forms a country-B subsidiary XB, that goes through the process of applying for the certs and setting up the payment server; or

- company X pays existing country-B-local payments company Y to use their payments server to do the money-moving-around for them; or

- company X pays multinational payments company Z to provide them a country-agnostic interface for moving money around; when requested by X to perform a country-B transaction, Z will then either signal the servers of their existing country-B-local subsidiary ZB — or will pass the message to their country-B-transaction-partner Y — to do the real money-moving around.

Anything that seemingly doesn't have one of these three fundamental shapes — but which still results in a company in country-A moving around money in a country-B bank account — is just window-dressing on top of underlying infrastructure that does have one of these shapes.