There is no requirement in Oregon’s statutes that the minutes be signed by anyone. However your policies may specify a process or procedure for signing by the board secretary, superintendent or board chair. The policy is usually included in policy BCB or policy BDDG.
In the book “Robert’s Rules of Order Newly Revised” the authors recommend that the minutes be signed by the secretary and, if the assembly wishes, by the president.
If there is no requirement in your district’s policy then your board can set up whatever procedure works best. Past practice (history) may also play a role here and you may want to ask how things have been done in the past.