For reference, Basic Attribute refers to the top row on your Attributes menu (Data, Vaccine, Virus, None, and Unknown). Natural Attribute refers to everything else (Fire, Ice, Water, etc.)
To put it simply, Natural Attributes are meaningless when it comes to deciding what you need to fight to earn more Attribute EXP. Contrary to the patch notes, fighting Ice will not get you Fire, Water will not get you Ice, etc. Instead, fight what your Basic Attribute is strong against. If your Digimon is a Fire/Vaccine, fight any Viruses you find. You Attribute EXP will be randomly distributed between your Va and Fire. You might get only one for awhile before you get the other, again it's randomly distributed.
For example, I fought WaruMonzaemon (Dark/Virus) with my Peckmon (Wind/Vaccine) and got EXP for both my Va and Wind Attributes. I then digivolved it to Crowmon (Dark/Vaccine) and still kept getting EXP for both Va and Dark. However, fighting Blossomon (Wood/Data) did not get me any EXP for Peckmon, despite the fact that Wind beats Wood.
The system was very poorly explained in the patch notes and is also still full of bugs. Repeated testing with Lopmon and it's champion evolutions has given me no consistent results, leading me to the conclusion that this particular Digimon is very bugged. So bear in mind that it still might just flat-out not work. I have a theory that Digimon whose Basic Attribute changes at different stages are bugged.
If anyone has any information conflicting with my findings, feel free to post. My consistent results were limited to Peckmon/Crowmon after all, so if you want to test this method with other species to see if you get the same results that would be cool.