To complement the other answers, and to explain the confusion:
Originally, there were two completely separate services:
- Eurostar served the routes between London and Paris or Brussels (then extended to Amsterdam), but not Paris to Brussels.
- Thalys served the Paris-Brussels-Amsterdam and Paris-Brussels-Cologne routes.
Recently, both services were merged into a single company (eurostar), but the services keep their separate operations:
- Trains to/from London require going through passport control (with so-called “juxtaposed controls”, where you go through the passport control of the destination before boarding rather than on arrival), and airport-like security checks
- All other trains require none of that
So, when you read about “Eurostar”, you are very likely to read about the services to/from London (because for decades those were the only services with that name), rather than the other services, which leads to the confusion.
There used to be a few complications with some trains between Brussels and London stopping in Lille, not sure if those still exist, they were a headache for Eurostar.