*Synonyms or related words for this sense of police dog
Collins dictionary

Webster
police dog

I understand your analogy.  However, in many areas a Recruit graduating the Academy still undergoes further training on the street.  After this field training, which can last several months they can work independently as a "Police Officer."   They are on a "probationary" period for up to 18 months.  A recruit graduating the Academy has training but no real world or "street" experience.  They may be "certified" Police Officers, but veteran Officers would never let them go on even a routine call by themselves.  They are "shadowed" for many months, until it is decided they are capable of handling the responsibilities of the job.  Some recruits graduate the Academy but fail to pass the "field training" stage and are not "cut loose" but resign or are terminated.  IMO, you are not really a "Police Officer" when you graduate the Academy, nor are you ready to be a cop.  Certifications and college degrees do not help at all when chasing a bad guy at 2 AM.  On the job training, learned skills, proficiency under stress and experience are what count. 

If a dog leaves police academy and is certified he is a "police" dog at that moment , even so he never served on the street.

  
Now, we are discussing different things.  Yes, I agree at that moment, it all hinges on what happens after that moment.  Dogs trained in a Police K-9 Academy and pass the certification, I would consider a "certified dog" expecting he has certified with his handler and is preparing to hit the street.  A dog that is trained on a bite suit, does some tracking, obedience and building searches I would not consider a trained Police dog.  I would consider that a well prepared dog to go to the Police K-9 Academy.  I find it to be a big benefit to be able to test and select dogs that have good foundation training.  I think that is a plus, granted the training is good.  We have a vendor near me that does terrible training, if we purchase a dog from them it is usually a newer dog just imported.  To prevent hem form screwing the dog up to much.  If anyone is going to screw the dog up we prefer to do that ourselves. 

This is purely differing views and semantics.  I understand your viewpoint and definitions, there is no argument here.  That is how you see it and you are correct in your way of thinking.  My feeling is that a Police Certification is nothing more than a sport test, not much different than a  SchH trial.  For our certifications the dog does a track, similar to a SchH 3 track maybe a little shorter.  The dog does off lead obedience, similar to a SchH 1 obedience routine without the dumbbell.  But we use voice and hand signals as well.  The dog does do an agility course, ours is rather demanding.  Then we do aggression control or bitework.  A call off, gunfire during one part and two outs. We do a long attack under gunfire with an out, a clean pat down and handler protection.  Dog is recalled to the handler to the heel position.  Then we do a building search with a decoy.  Sounds pretty much like a sport routine.  I can certify my dog at any time, we certify yearly and my next certification is in January.  I have never trained for a certification and will not train to pass my next certification.  Certifying is an easy day and really is the absolute minimum I expect from the dogs in our unit.  Any handler or dog that has issues with certifying has training issues that need to be corrected. 

Half of my post got cut off....