How do you put S3 on a civilian resume?

I looked at the headline and not your comment. Quantify everything. Operations Manager and supervisor of 5 individuals, responsible for planing daily operations which support 300 members in the organization. Mine looks like this: Managed 7K personnel logistics operation; responsible for coordinating the daily operational actions for 5 internal support companies and 5 deputy operating officers working in 29 separate locations providing logistical support to 32K customers.

Start with the basic concept you broke down for us: "I run training and operations for my Battalion..."

You're off to a better start than you think.

Operations and project manager.

Operation managers handle the day to day repetitive tasks project managers handle things that change constantly. ATRRS clerk is operations, training coordinator is project as you are constantly doing new tasks with identifiable start and end dates. I'd use operations or project as the title interchangeably depending on what type of company I am applying for.

The below definition is exactly the job description of an S3. https://www.pmi.org/learning/library/duties-effective-project-manager-5117

Project Manager job description as defined by the Project Management Institute "The project manager position on projects is critical; project managers must have full responsibility and accountability, must apply lessons learned, must define roles and responsibilitites, must lead project planning and tracking, must perform risk management, must apply best practices, must communicate to the project sponsor and team, must promote client involvement, must mentor, must promote good working relationships, and must make things happen."

I would translate it as. Oversaw the training and operations of over 300 personnel. Coordinating with upper level management to ensure that all training and mission goals were given the best opportunity for success.

State you were a planning and operations manager for an organization comprised of x soldiers, x millions of dollar equipment etc. just explain it without military jargon.

You can go to Indeed.com and type in the word Operations to get additional terms:
operations
operations management
operations supervisor
operations analyst
operations director
operations coordinator
operations associate
operations specialist
operations assistant

Use Operations Officer or Manager

It depends upon the job for which you're applying. I think "Operations Officer" is appropriate. Depending upon your level, you could also list "Operations Specialist." The title is less important than how you substantiate your experience and translate into something meaningful to civilians.

Depends on what job you are applying for , you need to hit key terms in your resume showing when and what skill sets qualify you for that job

You said it Operations and Training Supervisor. Sounds good to me. On the old, old, chart I had Artillery Operations, Inelligence NCO