Every coin you show in this thread has symbols. The elephant is a symbol for the country, the ships symbolize trade or military success, the peacocks beauty and grace. The peacock on the Southern Netherlands coin is in no way different from the peacock on the Burmese coin. Both are symbols.
On a different tack, the distinction between heraldic and what you call "natural" animals is vague. Things are clear as far as griffons and unicorns are concerned, but how about eagles? There are US coin with pretty natural eagles. The bald eagle (btw, never saw a hairy eagle) is both a symbol of the country and, as a heraldic eagle, the central device on the US CoA (as well as the logo of many government services.)
I will make some admittedly rather subjective distinctions here. I would distinguish between symbols and official symbols: the bald eagle is an official symbol of the USA; I could say that, for me, the European badger is a symbol of England, even though it is not an official one, and probably a few million Englishmen and women would agree with me. So let me say that the bald eagle is symbolic
of the USA, whilst the badger is merely emblematic
of England. As I said, a subjective distinction, but one that I think is easy to understand.
When an animal is a symbol of a country, its treatment on a coin will generally not be entirely naturalistic. You can see this on the US coin of which you provided an image: the eagle is in itself portrayed realistically enough - but look at those ethereal rays in the background, hinting that the eagle is somehow transcending its earthly nature and symbolising the USA itself.
The degree to which we perceive an animal as being portrayed realistically is again subjective. If an animal, which is the national symbol of that country, is portrayed on a country's coins, then we will tend to regard its depiction as more symbolic than realistic - whatever the artist's intent. So you are right - that adds a vagueness to my definitions: for you, that US eagle is fairly realistic; but for me, if I am collecting only "realistically portrayed" animals on coins, then that US coin will not end up in my collection. That is, however, an entirely subjective decision on my part.
To continue my definition, if a national symbol of a country is depicted against a natural background, such as the kiwi on this New Zealand 1933 florin, then the design is for me as much realistic as it is symbolic; in this case I regard the kiwi as more
realistic than symbolic, so it goes into my collection.