这 vs 这里, 我家 vs 我的家 to translate ”this is my home"?
这里 (here) refers to the 'location of the home' (an address, a place)
这 (this) refers to the 'physical home' (a house, a cardboard box) 这 can also be short for 这里 (here) in this context


As for 我家 vs. 我的家, you can see my answer to this question: Why no 的 in this sentence?


  The possessive '的' after a subject is usually kept, if it is a 'true possessive'(something you actually own) For example: The possessive 的 in 我的車 cannot be omitted and make it "我車"
  
  On the other hand, if the '的' is a 'relative possessive'(something you don't actually own) can be omitted. For example: you can omit the '的' in '我的母亲' and write 我母亲.
  
  More example:
  
  The possessive 的 in 这是我的太太; 他是我的女儿; 你的国家; 你的公司 can be omitted. You can write 这是我太太; 他是我女儿; 你国家; 你公司 (if you actually own the country or the company, then the possessive 的 should be kept)
  
  我(的)妈妈的房子很高。 can be reduced to: 我妈妈的房子很高。
  
  我(的)牛(的)角很长。 cannot be reduced to: 我牛的角很长。 or 我的牛角很长。
  
  (you possess the cow and the cow possesses the horns, neither can be omitted.)


Basically your answer used "这里" (here) instead of  "这" (this) is not a literal translation of the English sentence "This is my home". 


我家 can mean either "you live there and own the place" or "you live there but do not own the place (you could have rented it)
我的家 implies "you live there and own the place" or "you live there and you belongs to the family that own the place. In This case "我的家" implies "我(们)的家"

"这里是我的家" is not a wrong translation for "this is my home" if you are referring to a place you or your family owns but it can still be reduced to "这是我家"