:(Aside: it might be a good idea to register) You are correct on two main issues: (a) reference to power in the definition of decibel is misleading (restrictive); (b) the decibel is simply a mapping from a logarithmic to a linear scale.  However, your calculations are incorrect since the logarithm is defined only for dimensionless quantities, so 'log(y Watts)' is meaningless. 
 :Here is the problem with explaining the decibel satisfactorily: any definition of the decibel based on its historical roots (instead of mathematical "good practice") will remain troublesome.  A cleanup yields the following. 
 :: (a) Define B (bel) as a function such that x B =  for any positive number x.  Define d (deci) for any x by x d = x/10.  Define dB (decibel) by x dB = (x d)B, hence x dB = .  Example: 20 dB = 100. Similarly x B' =  and correspondingly dB' (example: 20 dB' = 10) for purposes shown next. 
 :: (b) Define dBW by x dBW = (x dB)W (example: 20 dBW = 100 W) and dBV by x dBV = (x dB')V (example: 20 dBV = 10 V).  Similarly, x dBW = ((x dB))W and so on. 
 :Done.  Insofar as a definition (which is always a convention) can be called "correct", the criterion is conceptual clarity plus convenience and safety in calculation.  You can verify that the definitions (a) and (b) also work out nicely and correctly for your examples.