Beats are places where fish can be found.
Fisheries are places specifically organized for fishing.

http://www.fishing-uk-scotland.com/htm/salmon-fishing-beats....
So a beat is a stretch of the river where the fishing rights are owned by the landowner of the land that the river runs adjacent to.

https://www.peche-jura.com/aappma-sirod-la-truite-du-val-de-...
Lot de pêche de 1ère catégorie domaine privé

High confidence for poste, low for lieux.

An English friend of mine used to own a carp-fishing lake in France.
A spot is any place from which one fishes and can be free, on rivers, but a 'beat', when you make your living from a fishing lake, is the 'spot' you rent. Large lakes can have several clients fishing at the same time, from their allocated 'beats' or postes (de pêche).

"Beat. A term used to describe an area of waterside bank on either a river or stillwater lake, that is allocated to one or more fishermen for their exclusive use over a time period."
"Spot : POSTE de pêche"
"Plan d'eau équipé d'un POSTE de pêche pour personnes à mobilité réduite, labélisé"
"Vous pourrez y pêcher:
• du POSTE situé à quelques mètres du chalet, tout en profitant de votre famille
• des différents POSTES de pêche aménagés autour de l'étang où vous pourrez, à volonté, installer votre Biwy [sic!]."
"L’étang Camalo dispose de 3 POSTES, dont un poste VIP avec chalet aménagé tout confort. Ces postes sont suffisamment larges (2 hectares par poste) pour permettre une pêche agréable et en toute tranquillité."

As for 'fisheries', I am guessing that it is used to cover any type of fishing space, be it a lake, a river, seashore, possibly even from a boat at sea. I am also assuming it concerns the rental of such places, not professional fishermen on their own boats or industrial fishing (factory ships).
I don't know which country this English comes from or what the law is regarding private ownership of beats and 'fisheries'. In France, I think river banks and shorelines are deemed to be public and can be fished for free, but lakes can be privately owned and thus fees applicable.






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Note added at 1 day 16 hrs (2022-12-31 10:23:49 GMT)
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Ah, je pense que le texte nous vient d'Ecosse: https://fishingpassport2.uk.w3pcloud.com/about/

"Scotland also has a range of PRIVATELY STOCKED FISHERIES, so if you are looking to teach the kids or you are looking for somewhere with facilities and less wild a PRIVATE FISHERY may be a perfect suit. PRIVATELY STOCKED FISHERIES will cost a bit more but will generally have toilets, car parking, food and drinks available and fishing equipment for hire."
"Raemoir Trout Fishery was originally constructed as a PRIVATE FISHERY on a single loch of approx 2 acres in 1995. The fishery opened as a PUBLIC TROUT FISHERY in 1997 and is now a Troutmasters Competition Water. Located one mile from Banchory, the FISHERY has 3 lochs and a bait pond covering about 6 acres within a naturalised 11 acre site."

Dans les exemples qui suivent on pourrait facilement traduire lieu de pêche par 'fishery', tel que ce mot est employé ce-dessus.

"Découvrez les spots [ ! ] de pêche à la carpe près de chez vous ou à l’autre bout de la France, profitez des équipements de confort, des services d’hébergement, de restauration, que les gestionnaires propriétaires d’étangs mettent à votre disposition sur leur LIEU DE PÊCHE"
"En terme de matériel et d'appâts, un pêcheur doit se soumettre à la réglementation du LIEU DE PÊCHE en eau close, établie par le propriétaire (hameçon sans ardillon par exemple)."
"L'étang des Persats est un LIEU DE PÊCHE à la carpe à louer en exclusivité, qui appartient au domaine de Lodgingcarp"
"L'étang du Soleil est un LIEU DE PÊCHE à la carpe situé dans l'Allier (03), à Audes, à proximité de Montluçon."

Mon niveau de confiance grimpe ...

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Note added at 1 day 19 hrs (2022-12-31 14:08:14 GMT)
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I've done some more research, and it appears pêcherie can indeed refer to a lake or a stretch of waterfront on lakes, rivers, or oceans where people can fish. Still, I had difficulty coming to grips with 'fishery' meaning anything other than vast sweeps of ocean, fish farms, or the practice of fishing at the industrial scale (meanings also conveyed by pêcherie), so, as a non-angler and for laypersons, I prefer lieux de pêche.