勤める, 務める, and 努める: employment, roles, and effort
Distinguish employment, performing a role, and making an effort with the shared reading つとめる, comparing に for an employer with を for a role.
Which organization employs someone
姉は銀行に勤めています。 says that the older sister works for a bank. に with 勤める, read つとめる, identifies the employing organization.
三年間この会社に勤めました。 says that someone worked for this company for three years. It gives a past employment period; further context is needed to know whether it continues.
Perform an assigned role
会議で司会を務めました。 says that someone served as the meeting's moderator. 務める, read つとめる, takes a role or position as its object with を.
委員長を務めています。 says that someone serves as committee chair. It does not by itself establish employment by a particular company.
What someone strives for
毎日、節電に努めています。 says that someone makes an effort to save electricity every day. に with 努める, read つとめる, marks the goal or area of effort.
分かりやすく説明するように努めました。 says that someone tried to explain clearly. The effort does not establish that the other person understood everything.
What do the spellings distinguish in 会社に勤める and 司会を務める?
Show the answer and why
勤める describes employment at a company; 務める describes performing the role of moderator. Making an effort uses 努める.