It depends on what you want to measure. If you only care about how tall the cake gets, then you can just measure it from the tallest point. If you want to measure how much space the cake takes up, you would want to measure the volume. Which method you use depends on how accurate you want to be. In general, when you have something irregularly shaped like a cake, the more accurate you want your calculation to be, the harder it is to measure. The simplest thing I can think of would be to approximate the cake as either a cylinder (if you’re baking in a round pan) or a rectangular prism (for a rectangular pan). Then you only have to take a few measurements.

For the case of a cylinder you would need the height and diameter of the cake. For the case of the rectangular pan you would need the height, width, and length of the cake. Then you can look up the formula for the volume of a cylinder or rectangle and plug in the numbers to find the volume. If you want to be even more accurate, you can use something really complicated like the authors in the paper you linked do. It sounds like they tried to get height measurements at many different points in the cake, and then construct a model of the top of the cake and did some calculus to calculate the volume of the cake based on that. The reason they have to do all this work is because there’s not a simple formula for the volume of really bumpy, wiggly, or irregular shapes (like the top of a cake!)

However, if you don’t care to eat the cake after you could get a pretty good estimate of the volume by seeing how much liquid the cake displaces when you submerge it in water. Since cakes are generally porous materials, some water will likely get into the cake — so it won’t be exact, but if you’re just comparing the volume of one cake to another it should be fine.

So it depends on what you want to measure. But if you’re interested in the amount of water that causes the cake to not rise at all, I would think just comparing the height at the tallest point would be OK. If you wanted to know how much water would produce the biggest cake, then I would say you should try to estimate the volume in one way or another.