That sounds pretty irresponsible of the owner...not even knowing what day their Yorkie went missing.
You have proved your integrity!
I have rescued dogs when neglect was obvious...food/water withheld and/or not provided, then the skin-and-bones dog turned out into the streets in such a state to fend for itself...and I don't care what the previous owner thinks under such conditions. My 'other' option would land them in legal trouble with fines and/or jail time, so they have received a huge gift in the rescue. If that turns out to be anywhere similar in the case with the Yorkie, especially if it is chipped, I'd present a similar 'alternative' and transfer ownership...I'd not want a chip to send the dog back there ever again in case something bad happened and it got away or was swiped, etc...and, regardless how careful and diligent we try to be, there are those occasions where things do go wrong.
I had a pair of Boxers show up in my kitchen once wearing ID collars, called the 'owner' but the dogs were also chipped and registered to a different owner. The 'owner' as identified by the collar had sold the dogs...it took a week to straighten out the 'ownership' issue before we agreed to release the dogs. The previous owner, as reflected by the tags, was a lawyer, so that helped because he guided us with what proof was necessary, etc., to avoid us releasing the dogs to anyone other than the 'rightful' current owner. The current owners were on vacation, on a cruise, and the dogs were being dog sat at their home where they tunneled under a 6' stockade fence ("no way those dogs are getting outta there"...famous last words...lol). They 'should' have disposed of the former ID tags and/or put current info on the collars, too...but did not. The dogs were well cared for, in great shape, very well trained and well mannered, slept with us in bed and we tried to fit in around them, and on the sofas, and were such sweetie pies.
Having been through that, and knowing of several other 'spaghetti ownership' snarls, I would want to make certain that my 'claim' was 100% valid and/or supportable. But, personally...having read the rest of the story...I kind of hope the Yorkie gets away again soon, and this time the owner admits they don't want her, have time for her...whatever is giving her the freedom to wander alone, and that they are only too willing to give her up to you! That may sound awful of me, but so far, the Yorkie's been lucky and been returned home...repeatedly. Who knows when the unlucky time will occur, and she's likely safer and more loved in your care.
However...
I am super thrilled
you have found a puppy
...looking forward to seeing pix.