How is "point" in geometry undefined? And What is a "mathematical definition"? How is "point" in geometry undefined? I mean, when we say "A point in geometry is a location. It has no size, i.e., no width, no length, and no depth," is it not a definition?  If it is not a definition, then how can we know whether some statement is definition or not? What are the characteristics of a definition in math?

I think your question is more about axiomatic systems in general. Maybe this analogy will help: Consider for example the axioms that govern set theory (called "ZFC"). The term "set" there is also undefined - even though we have some intuition about it.  From there we then go on to state various properties that sets have to obey.
More generally, when defining an axiomatic system (regardless if it's Euclidean geometry or ZFC set theory), you have "primitive notions" (points or lines resp. sets) and then you state property that relate the various primitive notions to each other.
The main point though is that while we use our intuition to help us find proofs and derive properties, on a formal level these are just manipulations of symbols that are not bound to our intuition. That allows us, if would like to do so, to replace the names of all primitive notions with other names.
Hilbert is famous for making such a remark, where he illustrates this idea taken to the extreme:  "One must be able to say at all times--instead of points, straight lines, and planes--tables, chairs, and beer mugs" (source: Provenance of Hilbert quote on table, chair, beer mug , where you can find also a bit of history).
