What are the halachik definitions of bittul torah?
I.e. some say that studying secular sciences qualifies as bittul torah while others say that it doesn't. How are they defining the halachik definition of bittul torah differently and on what basis?
The Gemara in Megillah 3a:
מכאן סמכו של בית רבי שמבטלין תלמוד תורה ובאין לשמוע מקרא מגילה
The house of Rebbi was mevatel Torah to come and hear the Megillah Much has been written about why the reading of the Megillah is considered bittul Torah, as the Megillah itself is seemingly a part of Torah.
The Yam Shel Shlomo explains that it is a qualitatively lower level of learning, and there is such a thing as ביטול תורה באיכות - qualitative bittul Torah. Others explain that the Mitzvah of reading the Megillah is meant to be an act of listening, not an act of learning.
Either way, even a mitzvah is called bittul Torah and requires justification (See the gemara there, which learns from a kal v'chomer that this particular Bittul Torah is acceptable). It seems that any act of not learning needs justification (I am making the assumption that the threshold of justification is more than "I was having fun;" otherwise the Gemara wouldn't very much need to tell us about reading Megillah being enough of a justification, nor to learn that it is a justification from a kal v'chomer).