The statute you cite is for municipally incorporated entities - cities and towns. A U.S. Senator can be expelled by the Senate, but not recalled: "As to removal by recall, the United States Constitution does not provide for nor authorize the recall of United States officers such as Senators, Representatives, or the President or Vice President, and thus no Member of Congress has ever been recalled in the history of the United States."
https://www.everycrsreport.com/reports/RL30016.html (https://crsreports.congress.gov/)


Ohio has no constitutional provision for officials elected to state or U.S. office. However, these states do have provisions for state elected officials:

Recall Provisions in State Constitutions and Statutes

Alaska – Const. Art. 11, §8; AS §15.45.470-710, 15.80.010, 29.26.240-350
Arizona - Const. Art. 8, §1-6; Ariz. Rev. Stat. §19-201 – 19-234
California – Const. Art. 2, §13-19; CA Election Code §11000-11386
Colorado – Const. Art. 21; Colo. Rev. Stat. §1-12-101 – 1-12-123, 31-4-501 – 31-4-505, 32-1-906 – 32-1-915
Georgia – Const. Art. 2, §2.4; Ga. Code §21-4-1 et seq.
Idaho – Const. Art. 6, §6; Idaho Code §34-1701 – 34-1715
Illinois - Const. Art. 3. §7
Kansas – Const. Art. 4, §3; KSA §25-4301 – 25-4331
Louisiana – Const. Art. 10, §26; La. Stats. Ann. §18:1300.1 – 18:1300.17
Michigan – Const. Art. 2, §8; Mich. Election Law §168.951 – 168.977
Minnesota – Const. Art. 8, §6; Minn. Stat. Ann. §211C.01 et seq.
Montana – Mont. Code § 2-16-601 – 2-16-635
Nevada – Const. Art. 2, §9; Nev. Rev. Stat. Ch. 294A.006, Ch. 306, Ch. 539.163 – 539.185
New Jersey – Const. Art. 1, §2(b); NJ Rev. Stat. Ann. § 19:27A-1 – 19:27A-18
North Dakota – Const. Art. 3, §1 and 10; ND Century Code Ann. §16.1-01-09.1, 44-08-21
Oregon – Const. Art. 2, §18; Or. Rev. Stat. Ch. 249.865 – 249.877
Rhode Island – Const. Art. 4, §1
Virginia - Va. Code §24.2-233 – 24.2.238
Washington – Const. Art. 1, Sec. 33-34; Wash. Rev. Code §29A.56.110 – 29A.56.270
Wisconsin – Const. Art. 13, §12; Wis. Stat. Ann. §9.10
https://www.ncsl.org/elections-and-campaigns/recall-of-state-officials

Impeachment was attempted once and was passed in the House. The Senate voted to expel him and then voted not to hold the impeachment trial.