I don't think that is correct.
IANAL but to me...
40-7-109Arrest by private person — Grounds.
(a) A private person may arrest another:
(1) For a public offense committed in the arresting person's presence;(b) A private person who makes an arrest of another pursuant to the provisions of §§ 40-7-109 — 40-7-115 shall receive no arrest fee or compensation for the arrest.
(2) When the person arrested has committed a felony, although not in the arresting person's presence; or
(3) When a felony has been committed, and the arresting person has reasonable cause to believe that the person arrested committed the felony.
(a)(1) is saying that you can arrest someone for ANY violation of the law committed inyour presence, including a misdemeanor.
(a)(2) is saying you can arrest someone if you know they committed a felony, even if the felony did not occur in your presence.
(a)(3) is saying you can arrest someone if you know a felony has been committed and you have reasonable cause to believe the person you want to arrest committed the felony.
You may also want to see...
40-7-110 Arrest by private person — Time.
40-7-111 Arrest by private person — Notice of grounds.
40-7-112 Arrest by private person — Notice of intention to make arrest — Use of force to enter dwelling house.
40-7-113 Disposition of person arrested by private person.
40-7-114 Pursuit after escape.
40-7-115 Breaking in to retake escaped prisoner.