My question is this: How did the Russian language end up having different endings in the phrases shown in the title of this post?
Now I will make some remarks to explain precisely what makes me puzzled.
I did some research and found that the form я студентка evolved from the form я/азъ есмь студентка, where есмь is the present tense first person singular form of быть and is no longer used in the Russian language (Source), so есмь is simply omitted nowadays.
Still, whilst была is followed by an object in the instrumental case, есмь is followed by an object in the nominative case, as can be seen in the following illustrative example below:
Азъ есмь Богъ, въ вѣкахъ предсказанный (Source)
Азъ есмь Софья. (Source)
и рече Игорь ко Асколду: «вы нѣста князя, ни роду княжа, нь азъ есмь князь, и мнѣ достоить княжити». (Source)
This means that even before есмь disappeared, it required a grammatical case different from the one required by был, despite есмь and был being one and the same verb taken in the present and past tenses, respectively.
Google shows that people also say я была студентка, but this variant is ~300 times less common than я была студенткой.
I wonder how one and the same verb быть can require one case in the present tense and a different case in the past tense.
P.S. Perhaps it is better to ask the question as follows: Why do people say кем он был, but never say кем он есть and prefer кто он есть (такой)?