In my understanding by comparing the Arizal's works to the Tanya, the animal soul = nefesh, the G-dly soul = neshamah. But what about ruach?
As elaborated in the source here, that is a misunderstanding. Both souls have all levels of Nefesh, Ruach, Neshama, Chaya and Yechida.
However, in the animal soul, Nefesh (the animation of basic life functions) is more predominant, and in the G-dly soul, Neshama (the intellectual aspects of the soul) is.
The five levels might be broken down as follows:
Nefesh = Thoughts, speech, and action (machshava dibbur umaaseh) Ruach = Emotive/Character Traits (middos) Neshama = Intellectual faculties (sechel)
Chaya = The very life of the person Yechida = The person in essence
The above is based off of Chassidus in general terms.
The problem is, that oftentimes different terms are used interchangeably to mean different things. For example, in the Tanya the two souls that a Jew possesses are referred to as NEFESH Elokis and NEFESH Behemis (G-dly and animalistic soul, respectively), but this has nothing to do with the level of "nefesh" as it is found as one of five levels of the soul.
Oftentimes the term "Neshama" will refer to the Jew's soul as a whole, while other times it will be talking about the Yechida, and so on..
So when making comparisons between the Arizal and the Alter Rebbe, you have to specify exactly in which source you are making the comparison in both works, so that one can see relative to that place what is being discussed.
I am not at all well versed in the writings of the Arizal, but I can only assume that when he writes "ruach" is referring to a specifically defined term in a given work, which may have any name in the Tanya, as the author decided call it...