> This also makes sense when you consider modulus operator. 37 % 7 gives 2 meaning 5 groups of 7 with 2 remaining in the last group.

Where is this coming from? The modulus is just integer division with a remainder. If you use your theory of interpreting this you can do it both ways.

Division is taking something and spliting it in X parts. By definition: the act of separating something into parts or groups

So when you are divinding by X you are separating the initial quantity into X equal parts.

35/7 means divide 35 in 7 equal parts.