Answer: Although I am not aware of an actively used distinction between the two types, the generic name "Amaryllis" is usually applied to a group of bulbs (Hippeastrum species included in Amaryllidacea) native to Central and South America. These include the large-flowered hybrids we find for sale for winter forcing and many of these are grown in the Netherlands. I suppose this would be the "Dutch" amaryllis you are thinking of.
As it happens, there is also an actual Amaryllis genus (also included in Amaryllidacea) native to South Africa. This would include another not very cold hardy bulb commonly known as the Belladonna lily. This then might be your African amaryllis.
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