What is the major reason why fungus-like protists are not fungi?

Fungal-like protists includes parasitic and predatory molds, water molds (Class Oomycota), and terrestrial slime molds (Myxomycota). 
These both move and lack the chitin that helps define true fungi. 
These fungal/amoeboid Myxomycota have a motile life stage fungi do not. However they do produce fruiting bodies to reproduce as do some fungi. Slime molds have an amoeboid phagotrophic (feeding) phase of their life cycle and use their pseudopodia to move and eat. In their mobile life stage they lack cell walls. 
Oomycota are aquatic fungi-like protists. They are water molds & plant parasites known as white rusts, and downy mildews. They have cell walls composed of cellulose, not chitin like true fungi. 
The sexual reproduction has heterogamous sex cells and asexual reproduction has motile spores (zoospores swim not sperm). The zoospores, with two flagella, are their asexual means of dispersal in water and damp soil.
Fungi have both sexual reproduction and asexual budding in yeast (that is not binary fission like bacteria). 
They are not capable of independent movement like an animal; they are sessile like a plant. 
Fungi are heterotrophic and use external enzymatic digestion of food stomach they digest their food outside their bodies and absorb the digested nutrients, this makes them osmotrophs instead of phagotrophs, which engulf their food.
Hope This helped! :D

The major, simple reason is that they can move while mushrooms cannot.

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