There are multiple different things that are being conflated here. This explanation is all US-based, and I’m more familiar with civilian trials than military. I may not be using the exact correct legal term in all cases (especially for military-specific stuff), but the basic idea should be right.
A confession can be done at any time to any person. I can confess to a crime to a family member, the police, a therapist, some dude I’m trying to impress, a reporter, or pretty much anyone. It’s just speech, may or may not be usable in court, and may or may not be taken seriously. “Acceptance” of a confession isn’t a particularly meaningful term.
A plea is telling a judge whether you accept the charges against you (guilty), require the state to prove them (not guilty) or something else like “I do not admit that I did it but do not contest being punished for the charges” (nolo contendere). Guilty and not guilty are always available, the ‘something else’ options vary a lot in different jurisdictions.
A judge must always accept a not-guilty plea, but for a guilty plea the judge will usually make sure the defendant realizes what he’s doing in a major case before accepting it. Pleading guilty on it’s own means that you don’t get a trial, and your sentence is entirely up to the judge and the law (or whatever authority handles sentencing), with the prosecutor typically asking for the harshest sentence available.
A plea bargain requires the prosecution and defense to agree on terms. The terms typically include a confession and guilty plea (or one of the special pleas) from the defendant, with a limitation on charges and sentence from the prosecution. (The judge also has to approve the plea bargain in the end, but usually only disapprove one in unusual circumstances). What the defendant agrees to confess as of a plea bargain is what they’re calling a ‘stipulation of facts’, and for major cases it will be done like a script, where the prosecution and defense lay out in writing what the defendant will confess to, then the defendant reads it out in court. If the defendant refuses to fulfil the conditions (say he gets to court and doesn’t want to say ‘I stabbed her’), then it typically voids the whole agreement. While plea bargains are common, it’s not required for a prosecutor to offer them, or for a prosecutor to act like one is in place if one isn’t agreed to.
So what’s happening is that Begdahl is going to the judge and saying “I want to plead guilty,” without any kind of plea bargain in place. Which means his sentence is entirely up to the judge, and could be anything up to life in prison since one of the charges is desertion. This is unusual for someone to do, because typically you either force the prosecutor to go to trial where you can defend yourself, either on real merits of the case or technical issues, or get some kind of deal limiting punishment. For example, Begdahl might have tried to get a deal where he limited punishment to the five year sentence that the lesser charge carries.
In a civil case this is like you were arrested for beating a guy to death in a fight, the prosecutor charges you with manslaughter, and you go to the judge and say “I did it, go ahead and sentence me to anything up to 20 years in jail the charge carries”. There would only be no ‘stipulation of facts’, that would only happen as part of a plea bargain. Confessing like this is unusual and bad for you as the defendant; typically you’d either plead not guilty, or reach a deal with the prosecutor. It’s possible if you were looking for a plea bargain (which Bergdahl has probably done) that you would go to the prosecutor and say “I’ll confess to ‘infliction of gross bodily harm’ and do five years for it if you drop the manslaughter charge and the chance of me doing 20”, but the prosecutor would say “No, you have to confess to ‘lesser manslaughter’ and do 10 years”, and you never reach an agreement. In that case if you go to the judge, there is no ‘stipulation of facts’ because you didn’t agree to anything, and the judge can sentence you to the full 20 years. You could also describe that as ‘the prosecutor didn’t accept my confession’ even though that’s not the legal terminology.