Where do rivers like the Rio de la Plata end and the ocean begin?
In the case of Río de la Plata, part of it is history and politics, and part of it is oceanography.

Most of Argentina and Uruguay considers Río de la Plata as a river (thus, the name, río) and as such it is the widest river in the world (maximum width >200km). Río de la Plata is formed as the confluence of the Paraná and Uruguay rivers and results in a pretty wide estuary with Montevideo (Uruguay) in the northern shore. 

As can be seen in the following satellite image, the combined discharge from the Paraná and Uruguay rivers creates a current that flows into the Atlantic Ocean and that is visible under calm wind conditions and high discharge periods.


Source: https://eoimages.gsfc.nasa.gov/

Spanish and Portuguese explorers in the XVI century described Rio de la Plata as a "Mar Dulce" (freshwater sea). The Italian explorer Sebastiano Caboto explored the area and traded silver with the local Guaraní tribes and thus gave it the name of "River of Silver".

In practice, the system is best described as an estuary: neither river nor ocean, but an area of freshwater and saltwater mixing. In the area of Montevideo, the salinity in spring is around 10 g/kg (way lower than ocean salinities ~30-35g/kg), but it can be higher in periods of low discharge. In the area of Buenos Aires, salinities below 5 g/kg are commonly found and thus, it makes sense to call it a river. Most oceanographers agree that the system is a salt-wedge estuary, with freshwater flowing near the surface toward the ocean and saltwater intruding landward near the bottom of the estuary.


Salinity of the Rio de la Plata: Source: Guerrero, R.A., Lasta, C., Acha, E.M., Mianzan, H., Framiñan, M., 1997. Hydrographic Atlas of the Río de la Plata. CARP-INIDEP, Buenos Aires, Montevideo.