There are two different teachings about what a "Daniel Fast" is.
Jentezen Franklin teaches about a "Daniel Fast" based on Daniel 1:8-12. This picks up where Daniel and his friends were taken into captivity, but selected as strong young men to be trained and educated as leaders in Babylon. Daniel and friends committed not to defile themselves with the food from the king's table, but to only eat vegetables and drink water.
Susan Gregory has written a book called, The Daniel Fast, based on Daniel 10:2-3. Here, Daniel committed himself to "eat no pleasant food," for three weeks. For Daniel, the food to be avoided was meat and wine. Though there are no specific foods listed that should be eaten, Gregory refers to vegetables (including fruit) and water. This seems to hearken back to the Daniel 1 passage.
Question: The passage in Daniel chapter 1 is really not about a "fast", is it? Daniel doesn't want to defile himself by eating the food from the king's table at all. This is a dietary decision Daniel is making for life, not a limited time of fasting. He asks Ashpenaz, the king's official, to test them for ten days and see what difference his dietary choice makes, but his plan is to keep eating like that for life.
Do you think it matters that Daniel 1 isn't referring to a "fast" when someone teaches that, anyway? Have you ever done a "Daniel Fast"? Like the "Trinity" or the "Rapture", the term "Daniel Fast" is never used in scripture. Does that matter?