I've had to write up both actors and crew before. It was always within a corporation so there was a set process followed within company Human Resources, however the basics are fairly universal. It should be on a separate form and should not be made public. In a theatrical setting I'd recommend it stay between yourself, the offending actor, and your immediate superior. If that person decides to share it with others, that's their business.
As far as what to put in the form the basics are a description of the incident, a statement by the offending employee if they choose to make one, and corrective actions to be taken to ensure a repeat does not occur, also date/time/place and all that great stuff that we SMs love to put everywhere
. When presenting to the employee it's best to be in a private setting. Explain to the employee why they are receiving disciplinary action, explain that the behavior he/she exhibited is not acceptable and explain why it is unacceptable. Let the employee read your warning, offer them the chance to write a statement, and after they have done so they should sign the form acknowledging that they understand. You should also sign the form in front of them.
If you anticipate any potential conflict it can be helpful to have your superior there. It can help ease tension between you and the employee, reassuring them that it's the company's view on the incident and not your personal view. It can also be beneficial on your end to have a witness present.
For situations below where you deem a warning is not required, I've found it's still a good idea to document the incident. If a future incident happens it's helpful to have a paper trail. In these cases I'd still include the offending employee on the discussion, and inform them that the incident has been documented.
It's not fun to do, but as a manager it's part of the job.
I hope you find this helpful!