I would appreciate it very much if you could recommend how we will quantify cleanliness criteria of a vessel and who has the final authority to say YES or NO?
As bimr has pointed out, this depends on many factors:
1) Application/service of the vessel. Can this particular application tolerate and up to what extent (size, amount, origin) of foreign material that is supposed to be cleaned before startup?
2) Will the residual dirt/debris/impurities accumulate in the system, or it will be flushed (and removed) somewhere downstream?
3) Nature of impurities: corrosive, erosive, incompatible with process fluid, or not?
4) Any reasonable means of removing the impurity down to the desired/specified level? If not, maybe you'll simply have to live with whatever you can't remove from the system.
In food and pharma practice there are methods for validating a cleaning procedure (using tracer chemicals).
Once you have qualified then you know that the process works.