There are two common species of Carassius in Europe:
[1] Crucian Carp / Gold carassius / Round carassius / Common carassius = Carassius carassius
Carassius: Latinization of karass, karausche, European crucian carp Actinopterygii (ray-finned fishes) > Cypriniformes (Carps) > Cyprinidae (Minnows or carps) > Cyprininae Max length 64 cm TL male/unsexed; common length : 15 cm TL male/unsexed; max. published weight: 3 kg; max. reported age: 10 years.
FWR 55-64 (80?) cm; 4-5 (6?) kg; 30 years. Freshwat.; brackish; demersal; potamodromous; depth 5 - ? m Temperate 2 - 22°C
69°N-35°N
10°W - 169°E Eurasia: North, Baltic, White, Barents, Black & Caspian Sea basins; Aegean Sea basin only in Maritza drainage; eastward to Kolyma drainage (Siberia); westward to Rhine & eastern drainages of England. Absent from North Sea basin in Sweden & Norway. In Baltic basin north to about 66°N. Widely introduced to Italy, England & France but possibly often confused with Carassius gibelio. At least one country reports adverse ecological impact after introduction. Adults occur in shallow ponds, lakes rich in vegetation & slow moving rivers. They burrow in mud in the drought or during winter. Usually restricted to densely vegetated backwaters & oxbows of lowland rivers. Can survive organic pollutants, cold, high temps & very low O2 conc. during summer & under ice cover. Feeds all day but mainly at night on plankton, benthic invertebrates, plant materials & detritus. Usually does not occur in waters with rich ichthyofauna & abundant predatory species, but very abundant in the absence of other fish species. Spawns in dense submerged vegetation. There is a gradual but continuing extirpation in many water bodies, esp. in Danube drainage & central Europe, possibly to due competition with introduced Carassius gibelio. Fisheries: highly com; a/c: com; game; aquar.: com. Least Concern
[2] Silver Crucian Carp / Giebel / Prussian Carp / Silver carassius = Carassius gibelio
Carassius: Latinization of karass, karausche, European crucian carp Actinopterygii (ray-finned fishes) > Cypriniformes (Carps) > Cyprinidae (Minnows or carps) > Cyprininae Max 35 cm SL male/unsexed; com. 20 cm TL; max. publ. w. 3 kg; max. rep. age: 10 years.
FWR 58 cm; 5 kg; 30 years. Freshwat.; brackish; benthopelagic; pH 7.1 - 7.5; dH 12 - ?; potamodromous Temperate 10 - 20°C;
62°N - 35°N,
10°W - 155°E Eurasia: usually considered native from central Europe to Siberia or introduced to Europe from eastern Asia. Clear & definite data on original distribution in Europe are not available due to introduction, confusion with Carassius auratus & complex reproduction. At present, widely distributed & commonly stocked together with Cyprinus carpio which is transported t/o Europe. Inhabits a wide variety of still water bodies & lowland rivers, usually associated with submerged vegetation or regular flooding. Can strongly tolerate low O2 concentrations & pollution. Lake dwelling individuals move into river mouths to avoid low O2 water in winter. Feeding larvae & juv. occur in high-complexity habitats as reed belts. Feeds on plankton, benthic invertebrates, plant material & detritus. Spawns in shallow, warm shores on submerged vegetation. Able to reproduce from unfertilized eggs (gynogenesis). Eastern European or wild form of the goldfish. Fisheries: minor com. Not Eval.
[3] Goldfish = Carassius auratus auratus / Carassius auratus
Carassius: Latinization of , karass, karausche, European crucian carp Actinopterygii (ray-finned fishes) > Cypriniformes (Carps) > Cyprinidae (Minnows or carps) > Cyprininae Max 32 cm SL m./u.; com. 10 cm TL; 41 years. 40 cm, 2.3 kg (Pract. Fishkeep. 2010).
FWR 50-62 cm; 3-4 (5?) kg; 50+ years. Freshwater benthopelagic; pH 6.0 - 8.0; dH 5 - 19; potamodromous; depth ? - 20 m Subtropical ? - 41°C
53°N-22°N Asia: central Asia & China, & Japan. Introduced t/o the world. Asian form of the goldfish. Several countries report adverse ecological impact after introduction. Inhabit rivers, lakes, ponds & ditches with stagnant or slow-flowing water. Occur in eutrophic waters, well vegetated ponds & canals. Live better in cold water. Feed mainly on plankton, benthic invertebrates, plant material & detritus. Goldfish lay eggs on submerged vegetation. Females spawn multiple times during the spawning period. Oviparous, with pelagic larvae. Max. recorded salinity is 17 ppt, but unable to withstand prolonged exposure >15 ppt. Used as an experimental species. Edible but rarely eaten. Aquar. keeping: in groups of 5 or more; min. tank size 1 m. Not Eval.
Perhaps Japan has their own species of common Carassius. Crucian carp are Carasius carassius from northern Europe, also a favorite and most spread and fished for fish in my home country. I fished all my younger years for them and carp too.
Perhaps I am wrong but i think goldfish is just a selectively bred for 5000 years Carassius carassius that became C. auratus, or maybe some very closely species out of Eastern Asia, likely the gibelio?
Koi BTW are a selectively bred far-eastern carp species Cyprinus rubrofuscus or Amur river carp, not the common carp Cyprinus carpio.