No. Any Ohio city, with an income tax, is allowed to have withholding taken out of your pay while you worked in that city. Your employer is REQUIRED to do that withholding. He is not required to withhold for your residence city.
Employers are only required to withhold for the work city, although many will voluntarily withhold for the resident city as well. Employees of most big companies, in Ohio, seldom owe any city tax, at tax time (but they still have to file a resident city return), because the employer holds out exactly the right amount for both work and resident city, taking into account the credit.
Yes, many people pay have to both cities, although "double" is rare. You will not be paying Shaker Hts the full rate, after the 1% credit. But not all people have to pay twice, many cities allow their resident a full credit for what they paid the work city.
Yes, there may be a better way to administer the system. But, this is the one we have to live with. You're not the first one to have to find out about this the hard way.
