I'm not sure if "anyone has ever conducted a study" of what some scholars consider to be aggada and others do not. However, we should be wary of such black and white distinctions, and of assuming that just because a passage is quoted in the Ein Yaakov, R. Yaakov Chaviva thought it should fit squarely into the "aggada" box. There are passages that the Maharsha comments upon in both his chiddushei aggados as well as his chiddushei halakhos, because it is possible for a halakhic discussion to have aggadic/philosophical implications, and vice versa. Therefore, one cannot merely say that just because a passage was included in a list of aggados such as that of the Eyn Yaakov of Maharsha does it become non-halakhic.
The distinction between halakha and aggadah was one that existed at least from the time of the Mishnah and Gemara. There are many sources to this affect, such as Sanhedrin 67b, Sotah 40a, Bava Kama 60b and others. This distinction seems to have major importance considering that there's a principle thrown around by poskim here and there (see Tos. Yom Tov to Brachos 5:4 and Maharal below) that we shouldn't make legal implication from aggadah. To list all of the places where these lines are blurred, and how others have viewed them, would take a book's length discussion (though I'm unaware of any such book as of yet).
What I can do here is answer the title of the question, 'how do different scholars define aggada'. As far as I'm aware, everyone differentiates in pretty much the same way: halakha is anything that pertains directly to 'what to do', and aggada is everything else.
R. Shmuel ben Chofni Gaon, in his intro to Talmud (that was attributed to R. Shmuel Hanagid) states that "aggada is any explanation mentioned in the Talmud that does not explain a commandment". He goes on to give another difference between them, that halakhos were received by tradition, and aggados are the personal opinions of the chachamim. (This second difference seems to be incidental rather than definitional)
A very similar concept can be found in the Ramban's 'disputation', where he explains that the Jews have three types of books, the bible, the Talmud which is an explanation of the commandments in the Bible, and we have 'sermons', which were speeches that the Rabbis gave in public. It seems pretty clear that the Ramban is distinguishing between halakha and aggada, saying that halakha is anything that explains the commandments, and aggadah is everything else.
Maharal in Beer Hagoleh 6 also writes that there's no issue of having contradicting aggados because the whole principle of aggada is something that does not affect actionable commands. אבל באגדה שאין דין יוצא מזה אין להקשות ולסתור... ובשביל כך אמרו, אין למדין הלכה מתוך האגדה, כי לא נתברר על ידי קושיות ותשובות
Gr"a to Mishlei 25:21 compares halakha to fire and agada to water, inter alia saying that halakha pertains directly to actions (which are exacting, as fire has an immediate affect on what it combusts) and agada to mussar (which dissolves things slowly over time)
R. Hirsch in his intro to Horeb reverses what was said by R. Shmuel (above), saying that halakha is called 'shmaatsa', because the emphasis is that it was heard, student from teacher in a chain extending back to Sinai, as opposed to 'aggada' which was just 'הוגד', said, without its author having heard it from his teacher. Therefore, says R. Hirsch, halakhos are legally binding as actions and aggados are merely ideas.
R. Dessler (Michtav Meeliyahu vol 4 pg 353) makes the usual distinction, but has a very interesting conclusion: since the purpose of halakha is to tell us what to do, it's ok if we don't understand why we do it, as long as we do it, because it's function is 'to do'. Aggada, however, exists to be understood, and we therefore have an obligation to dwell on its meaning until we understand it, because it isn't something to be acted upon.
The questions of ambiguity appear when 1) aggadic concepts are used to drive halakhic conclusions, 2) when a passage in the gemara sounds clearly like a story, but nonetheless includes someone doing something that has halakhic implications, or 3) halakhos are recorded in the Gemara in a manner akin to aggados.
Tosfos in many places seem to take aggados seriously in terms of how to pasken halakha. Two instances that come to mind are Bava Basra 74a, where Tosfos conclude from a rather strange story that the dead should be buried while wearing their tzitzis. Another place is from Sanhedrin 74b which discusses the halakhic question of why Ester didn't allow herself to be killed, and Tosfos ask a question on the Gemara based on an aggada in Megilah. The Rashba, though, in Megilah 15a disagrees with Tosfos, and writes simply דברי אגדה הן ואין משיבין עליה. It sounds like the issue at hand is whether or not the difference between halakha and aggada can really be attributed to actionable statements vs. non-actionable statements.
For a different discussion (an exploration of what's presented as three differing opinions), see this shiur from vbm-torah.org: http://vbm-torah.org/archive/taggada/02taggada.htm