I'm keen on the idea of baking using "Baker's Math"... but, what ingredients do we count when trying to figure out the hydration level of a dough? Do we count just water/liquid? Or, do we count things like sour cream, butter, milk, eggs?
The term is not really defined that way. In the strict sense, it only has meaning for lean doughs - made of only flour, water, yeast and salt, eventually preferments or later mixins like seeds.
You can extend it to enriched doughs (those that have fat, milk, eggs, tomato juice, etc.) but then it is no longer sharply defined. Calculating the exact content of water (e.g. adding to the calculation 17 g of water for every 100 g of butter) are pretty useless, because the water bound in an emulsion or other kind of colloid doesn't behave the same way as free water when making a dough. So, a dough with 100 g water and 100 g butter will be extremely different from a dough with 117 g of water, and still pretty different from one with 117 g of water and 83 g of oil. So, there is no real formula any more to create an actual percentage.
The concept still stays somewhat useful in this case, but you have to "play it by ear". When you are trying to judge how soft a given enriched dough will be, you can start by calculating all the liquid and fat ingredients as if they were "hydration", which gives you a rough guide to compare it to other doughs, and then you have to make it and gain a tactile understanding of how it actually behaves.
It depends on the ingredients and how precise you want to be. Things like butter, milk, and eggs do include water and do affect the hydration of the dough.
The water content of butter varies widely, but 15% is in normal range in the US. So, 30 grams of butter might have around 4.5 grams of water in it. If you add that 30 grams of butter to 400 grams of flour, that's only about a 1% change in hydration. That likely isn't enough for most people to worry about. Although, some bakers might care enough to include it in their calculations.
On the other hand, ingredients like milk contain a lot of water and tend to be used in large quantities, so they really should be included.
Dough hydration is the amount of water in the dough. See: wikipedia
To calculate, you need to figure out the % water of each individual ingredient and then divide the total of all water in the recipe by the total weight.