Statement #1 : Galaxy gravity causes distortion of space.
Statement # 2: It is the distortion of space by Galaxy which causes Gravity."
I do not know which one is the right statement but both cannot be true together. GR description is leaning more towards statement # 2.
First, I believe you are confusing the concepts of "space" and "spacetime". To be honest I have seen times where even those who know the difference use the words in a confusing manner, but generally they are speaking to others who understand...
Though there could be a relativistic argument on the issue, space should most often not be thought of as curved and/or distorted by the presence of mass. Spacetime is a different animal and is a construct composed of more than one part, including, but perhaps not limited to, "space", "time" and even "energy". The balance of how those components of spacetime interact dynamically, both with each other and any local mass, generates what is referenced as spacetime curvature. That curvature describes the gravitational field...
So the first statement is accurate, "Galaxy gravity causes distortion of space.".., but unclear the way it is phrased. Distortion is a muddy word here. The mass of a galaxy generates a graviationnal field, that can be described as a curvature of spacetime. (Qualification, modern attempts to develope a model of quantum gravity, approch the generation of a gravitational field as an interaction between mass and spacetime, though spacetime is not the way that process is initially described.., and there is a modern interpretation of GR that does define the curvature of spacetime as the cause of gravitation. Both are theoretical, as in unproven.)
Your second statement, "It is the distortion of space by Galaxy which causes Gravity.".., could be thought of as consistent with the second part of the qualifying statement above, but remember it is theoretical and depends on accepting the underlying conceptual and theoretical model.
It is difficult to separate descriptions of spacetime curvature from gravitation, but that does not really mean that either one causes the other. You can think of the two as the same thing, in different words.., and still not know the fundamental why of, or how gravitation emerges from the presence of mass.
As far as I am concerned I cannot visualize any space curve distortion or time curve distortion or even spacetime distortion qualitatively. I can understand the space time distortion at the end boundary (if existing) of universe, but as far as any object is concerned (say Earth), then it is space all around it, how it can distort ? To me it appears to be a completely abstract concept, beyond my limited intelligence.
They are both abstract concepts, relative to everyday experience. It has really only been since the developement of high precision clocks and things like the GPS system that we have any practical examples of time dilation, what I believe you are referring to as time distortion... And we still have no direct evidence of length contraction. These two together play a large role in what is usually meant by curvature of spacetime, though spacetime cuvature gets even more complicated and involved than just time dilation and length contraction.