These are expressions you sometimes hear from people who know no better. They are usually trying to simply say that their puppy/dog is even smaller than the norm. If you hear it from a pet owner, just nod your head and admire their cute little dog.
If however you hear it from a supposed breeder, be instantly suspicious. These classifications do not exit. Any breeder, who implies any different, usually has more money as a motive for applying this term to their dog. These pups are usually the runts of the litter, and as such are more prone to ill health. The runt puppies may be possibly that size because of health problems.
Others in a liter though may live long healthy lives. It is the runt AKA “tea-cups” with health problems that can cause distress to new owners when they either die at a very early age or become a financial burden requiring constant veterinary care.
So in reality by wanting the tea-cup AKA “runt” what you’re asking for is the least hardy of the liter. The runt or least hardy puppy would be the most delicate or fragile of the liter. Your asking for less and hoping for the best. Potential buyers need to be aware of the many heath issues such a hydrocephalus or heart problems.
Within a litter, the sizes of individual puppies when adult can vary considerably from the very small to the very large, so even viewing the parents will not necessarily give an indication of the eventual size of a young pup. If you really want a very small dog, you are best to leave buying one until it is at least 4 months old and then even 6 months is better. At this time it should be clear as to their size and health status.
When buying a puppy, seek out a source registered with a governing body in your state (ex. NYS Dept. of Agriculture).