Nourishment is the benefit that food provides.
The words are not interchangeable.
We eat food.
We do not "eat nourishment". We get nourishment or take nourishment from the food we eat.
Nourishment is abstract. Food is the word for tangible, edible things :)
"Cook" focuses more narrowly on the act of applying heat and the transformation that the food undergoes because of the heat. Commenters on the question have properly noted that the more specific "bake" should be used instead of "cook" when the heat is applied in an oven (especially when the transformation is more than just increasing the temperature of the food).
"Make" is more general and covers other aspects of food preparation that do not necessarily involve the application of heat.
For example, you "make" a salad but you don't "cook" a salad, because there's no heat (usually). You might cook chicken as part of making a salad.
You "bake" a cake by putting it in the oven, and it is implied by context that you took other steps as well (such as mixing the ingredients, greasing the pan, frosting it, etc.) but if you "make" a cake those other steps are more explicitly stated and more emphasized.
In your examples, I would tend to use "make" more often unless "cooking" was about the only step in food preparation (e.g. "cook a steak" for some preparations). I would also use "cook" or "bake" if the heat component was the focus of my narrative (e.g. "I went to cook the pasta and discovered that my stove didn't work" or "I feel hot and tired because I've been baking bread all day").
One also hears people talking about how "accurately measuring ingredients tends to be more important in baking than in general cooking," because the ratios of one ingredient to another are important for the chemical reactions that take place while baking.
If you use "cook/bake" interchangeably with "make" when referring to food preparation with heat, or if you always use "make" for food preparation, the meaning will be quite clear*, and you don't need to worry about using the wrong one. The StackExchange Q&A site for chefs is even "cooking.stackexchange.com." However, if you want to understand the subtle distinction a bit more, I hope the answer above has helped.
(*) with very few exceptions, such as if you're a known sushi lover talking about "making fish for dinner" with an intended meal partner who strongly prefers that the fish be cooked; this is someone who might seek clarification.
Best Answer
Is your friend a native speaker of English? I suspect she might be mapping words from another language onto English. (Rarely is there ever a perfect fit.)
In East Asia, for example, rice is the most important staple, and in most areas some kind of rice or rice product is eaten with almost every meal. It is so central that the Chinese character 飯 indicating cooked rice can also mean food or meal in general, in Japanese as well as Chinese use. In Korean 밥먹자! (bab meokja!) translates literally as "let's eat the cooked rice!" but ordinarily just means "let's have something to eat!" or "let's dig in!"—even if you are having goulash or tacos.
The point is that historically, if you asked someone what she was going to have for dinner, she would never have answered "rice"— that would have been taken for granted. I have older Korean relatives who to this day will chuckle if I tell them I had rice— well, obviously. When they ask me what I had for dinner, they want to know what meat, stew, and 반찬 (banchan, side dishes) I had. This seems to correspond to the distinction your friend is making between the staple/core of the meal and its accompaniment.
None of this applies in English, however; in standard usage, the distinction your friend makes does not exist. Rice is simply one type of food. Declaring that food only refers to foods other than rice is unsupported either by common usage or by dictionaries. I do not know if her use is common, but anecdotally speaking, none of the people I know who might refer generically to food or a meal or a dish as 飯 or 밥 in another language ever use rice in that way in English.
Bread is the nearest Western equivalent. We still have many expressions involving bread in English, like breaking bread (sharing a meal) with someone, something being your bread and butter (essential or basic), or knowing which side your bread is buttered on (recognizing a benefactor or source of advantage). Still, I do not think bread ever attained the cultural centrality in the English-speaking world that rice did in southern and eastern Asia, and I have never come across anyone who would try to declare that bread was in a special class of edibles. Bread is food, just like everything else you eat.