Mr. Rossmeissl's absolutely correct, and his answer is as true of Texas courts as, I presume, of California's.
Some of the caselaw makes a very peculiar distinction between "civil contempt of court" and "criminal contempt of court," but the difference doesn't depend on which type of COURT is involved. Typically, when what's being asking for is to force someone to do something -- like show up, or testify, or produce subpoenaed records -- that's considered "civil contempt of court." The person who's refusing to obey the court order -- called the "contemnor" in contempt proceedings (although he may also be "the defendant" or something else too) -- is said to "hold the keys to his own jail cell," because he can "purge" himself of contempt, and come back into compliance with the order, by simply agreeing to do what the court has ordered. "Criminal contempt," by contrast, is purely a function of punishment, one in which the judge says, in effect, "You broke my order, now regardless of whether you wish to comply in the future you're going to do 30 days' jail time," or the like. Either way, the punishment can be enhanced by fines. Regardless of whether you're in criminal district court, civil district court, probate court, county civil-court-at-law, etc., you could be held in either criminal or civil contempt because the kind of punishment sought determines the kind of contempt, not the underlying litigation or court.
Judges may, and sometimes do, hold parties or non-parties who've been subpoenaed in contempt "on their own motion" -- that is, without anyone first suggesting it to the judge. Or other times, a party may file a motion asking for someone to be held in contempt. That's what would typically happen in the situation you describe of a missed deposition: The party trying to secure your testimony in connection with a pending probate court case would have you subpoenaed, and then when you failed to appear at the appointed date, time, and place, that party would have the court reporter prepare a "certificate of non-appearance," which becomes the proof that you didn't comply. That party would take the certificate to the court and ask for an order compelling your attendance (and quite possibly also commanding you to pay the wasted attorneys' fees and expenses associated with your non-appearance, to make others whole); such an order would typically give you one last chance with another a time and date and place to appear, and upon proof that you didn't comply with THAT, then that's when the court is likely to issue a civil contempt order directing the sheriff or any law enforcement officer to arrest you and bring you directly to court to quite literally face the judge and account for yourself.
If this seems scary, it should. Contempt of court is a powerful remedy, one with which the appellate courts rarely can come to your rescue.
Fortunately, there is almost always a better path.
If your problem is just that you have a scheduling conflict, usually you can informally arrange with the lawyers -- starting with the one who's sought your deposition and had the subpoena issued and served on you -- to find another mutually convenient time, date, and place. The longer lead-time you give them, the more flexibility they're likely to have, so don't wait until the last minute to ask to reschedule.
If your testimony really is extremely unlikely to be relevant, or if the deposition is just being sought to harass or embarrass you, and if you can't persuade the lawyers just to drop the whole idea, then you can apply to the relevant court where the case is pending to ask for a "protective order." Such an order might quash (terminate) the deposition altogether, or it might specify a date, time, or place that the lawyers were unwilling to agree upon with you, or it might limit the subject-matters upon which the lawyers may inquire. Your chances of getting such a motion granted will be much, much better if you retain counsel to help you.
Generally, you seek to enforce the deposition in the court in which the action is pending, so if there is a probate court lawsuit, and a witness fails to attend the deposition as required, you would go to the probate judge for an order compelling compliance (or for an order finding the witness to be in contempt, if that is the remedy you seek). You should consult with an experienced probate litigation attorney for assistance.
This general response is not intended to create an attorney-client relationship.