Both are used in BrE too.In my dialect, AmE, they are both idiomatically correct and are used interchangeably.
I would usually say "I'm going to cook the dinner". I wonder if this is another Brit/Am difference? (I do all the cooking in our house, I like it too).They are not interchangeable for me!
I need to get home to cook dinner. I would NOT use "the" there unless we had already spoken about a dinner party or something that I had prepare.
After working in the office all day, I actually like to cook dinner. It's fun and relaxing for me. I would not use "the" there either. I would only use it in something like "I actually like to cook the dinner that my family has been looking forward to all day." (Because I'm such a great cook, I guess.)
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