I wouldn't be surprised at all if there were some commuter trains into Cleveland. Before 1900 Central and Central-affiliated lines operated commuter service into many of the major cities. The streetcar lines probably killed a lot of the smaller services around 1900 and then the double-whammy of increasing auto ownership and the Depression probably killed off many more in the 1930s.
As an example there were suburban passenger trains operated out of Cincinnati on the Big Four and labeled as such in public timetables.
Looking at an LS&MS August 1912
public timetable on the Canada Southern site (link to the homepage)
, shows several trains that were probably used by commuters (or could've been).
On the main line east of Cleveland the schedule shows:
Train 115 originating at Conneaut OH at 6:15 am and arriving Cleveland at 8:30 am. Operated Mon-Sat.
Train 116 departing Cleveland 4:30 pm and terminating at Ashtabula at 6:05 pm, Mon-Fri.
Train 118 departing Cleveland 12:50 pm and terminating at Ashtabula at 2:30 pm, Saturday only.
On the LS&MS main line west of Cleveland the schedule shows:
Train 126 Originating at Norwalk (TN&C Branch) at 6:05 am, arriving at the main line jct of Elyria at 7:00 am and arriving Cleveland at 7:45 am, Mon-Sat.
Train 127 Originating at Cleveland at 5:05 pm. leaving the main line at Elyria at 5:52 pm and terminating at Norwalk at 6:50 pm, Mon-Sat.
There was only one local station shown on Cleveland's west side (West Park) but four between Cleveland (MP 183) and Collinwood (MP 177). West Park was served in both directions by the Norwalk train and the east Cleveland stations were served by the Conneaut / Ashtabula train.
East 40th Street (MP 181)
East 55th Street (MP 180)
East 105th Street (MP 178)
Coits (MP 176)