But there's one thing that "doesn't add up" and it seems to be "logic" based...

if adding c to x where c>0 results in a horizontal shift to the left c units (aka horizontal shift of -c units)

and

multiplying c by x where c>1 results in a horizontal compression by a factor of c (aka horizontal compression of c units)

then observe the above statements in the parentheses.

c becomes -c (addition/subtraction)

c becomes c (multiplication/division).

This seems wrong.

If c becomes -c (addition/subtraction)

then c should becomes 1/c (multiplication/division).