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In OmniPlan you're able to set custom off-hours and overtime hours for each resource in the project. If you are a resource in the project, you could select yourself and mark a few hours in the day as off-hours if you had an upcoming doctor's appointment or something to that effect.

Once you did this and published your project, the change in availability would be updated on your colleagues' copy of the project if they were also set up to sync with you.

Having said that, the example you mentioned of moving a task to a new start date and having it appear on the new start date in your colleague's copy of OmniPlan is also true.