In my experience the civil terms "with prejudice" and "without prejudice" are not generally used with respect to orders of dismissal in criminal cases. As to whether the prosecution can refile a case that has been dismissed, that depends on the grounds for the dismissal. Often yes, but not always.
More facts are needed. If it is early on and there is no form of pretrial diversion, then you're correct they are not generally dismissed with prejudice. If you complete some form of pretrial diversion, probation, etc then it is dismissed with prejudice in effect because charges can never be filed based on this charge again. Post more info about what you're asking and more complete answers will be possible.
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Prosecutors generally have total control over whether a criminal case is filed, dismissed, or prosecuted. As long as they have admissible proof beyond a reasonable doubt, theoretically they may charge a person anytime within the statute of limitations, even after a dismissal. There may be other concerns for the prosecution, like constitutional speedy trial rights. But as a practical matter, once a prosecutor dismisses the case they move on and do not revisit it. They may dismiss for a tactical reason, but I find that rarely occurs.