That is my definition also.
Once you have measured something the observation is a scalar. If you perform some experiment and the number 7.43 pops out on your measuring device then no change of coordinate systems can possibly change that number to anything other than 7.43. Therefore, the number measured is a scalar.
It may be that you claim that 7.43 is a length and I disagree, but regardless of how we interpret the number in terms of physical quantities in our favorite coordinate system, we will agree that the number is the same. That makes it a scalar.