Write City of St. Louis when you mean the city, and county when you mean the county
The fastest way to sound like an out-of-town firm is to use St. Louis as though it named one place.
The City of St. Louis is an independent city. It sits outside St. Louis County, it has its own government, and the two operate under different rules. Almost nowhere else in the country is set up this way, and everyone who lives here knows it.
Search behavior follows the same split. A parent in Webster Groves or Kirkwood searches with the municipality or with the county. The bare metro name is what people from out of state type. A page written for the bare metro name matches the low-intent search and reads as generic to the person who does land on it.
The fix is unglamorous. Give the city and the county separate treatment in page titles, headings and service area copy, and say in plain terms where each matter would generally be filed. Local practice differs between the two, and the specifics are worth confirming with the relevant court or your own counsel before you publish them.
The payoff is not only ranking. A reader who sees you make the distinction decides you are actually from here, which is most of what the first screen of a legal website has to accomplish.