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Tables are finicky and tricky. And possibly buggy.
If you placed/pasted a table along with its attributes and then proceeded to dress it up with direct formatting and cell styles, you will soon have an unworkable mess.
Best bet, build a hierarchy of a Table Style that calls for 3 or more Cell Styles. Each Cell Style should call for a defined paragraph style. Build all these styles without using the Type tool inserted into any text/cell/table. In other words, imagine it. Place the table from Word or Excel, bringing it into InDesign without any formatting. Then apply the Table Style and clear overrides in the Cell Styles panel.
Look out: your multi-colored header is too complex to build into a Cell Style. It will require many cell styles, and you might therefore decide to not call for any header cell style for the header cells in the Table Style.
Making table and cell styles cleanly means no hidden corrupt counter-commands on how your table should behave.
Further, your example file's Table Style is not defining the Header Cell Style correctly; nor the Body Cell Style, either.
Okay, I see. The styles aren't as clean as they could be. I've never set one up as you described before.
One thing I did notice upon inspecting for counter-commands is that there are Keep Options for Cells. Apparently the row had its start option set to Next Text Column, which causes what's shown in the file to happen. This doesn't seem like something that can be set with a Cell Style though.