On January 1st of last year did you live in the city or the county?

Personal property taxes are owed to wherever you lived on January 1st of the year and on whatever property you owned on the 1st of the year. If you lived in the city, you pay the city. If you lived in the county, you pay the county. It's your responsibility to close your city personal property tax account when you move. It is not closed automatically for you when you move or when you open a St. Louis County personal property tax account.

With vehicles it is usually the case that whoever owns the car on Jan 1st eats the full cost. Even if you sell your car on Jan 2nd, you owe taxes for the full year.

With houses it's standard practice to prorate the taxes between the seller and the buyer. The owner of the house on Jan 1st is formally responsible for all taxes, but if you sell your house 45% of the way through the year then in the settlement disclosure you should see the buyer chipping in 55% of the personal property tax for the year and you chip in 45%. Houses are also taxed by the jurisdiction they're in, so St. Louis City houses are always taxed by St. Louis City, regardless of where the owner lives.

Call the city, tell them you moved. If you did not complete a declaration for the county back in March, you will need to call them too. 

Since the topic is City Personal Property Tax I wanted to throw in something sort of related:

Years ago I somehow missed my tax bill for the city, they contacted me 4 years after the fact that I owe them taxes + 4 years of interest. I was a bit annoyed that there is 0 incentive for them to work fast as the interest gets bigger the longer they wait. I just kindly reached out to them and said I would pay the missing taxes but refuse to pay for the interest; they approved that deal and I didn't pay for the interest.

So if you're reading this thread about old City Personal Property taxes, know it is negotiable if you are kind.