Once again, for those who do not understand: the term of a "paper warrant" is used in cases where the lead detective on the case often shows the person in question an actual written warrant on paper, yet does not enter it into the "system" or state/national database where any other law enforcement official could potentially act upon it.
In other words, it is often a bargaining chip in cases where the investigators are searching for better leads/information in hopes of solving a bigger problem issue. They want to be the ones with the power and control, and entering it into the system would do absolutely nothing in that regard.
In all actuality, I do think you have a point that it's either an active or an inactive warrant--especially if it has not been officially signed yet, so in that aspect I agree with you 100%; but there does exist such a thing often referred to as a "paper warrant." Here are 3 examples off of the top of my head:
1. In the state where I live (PA) we have constables who serve such warrants when someone is behind on their court costs, as the constable receives a percentage of the cost if he/she collects the money from the individual; however, it is rarely if ever entered into the main database for another law enforcement official to act upon, as they simply don't have the care, time, or resources to waste upon picking people up for something as little as a court cost violation. The warrant is certainly still an active warrant as the constable will often take the individual to jail, where he/she will serve the tradeoff of $40 dollars per day towards their fines, or remain there until someone pays the constable off.
2. Another example would be for someone who has been picked up on a drug charge. As we know the drug violations are a "chain" of sorts where the police/investigator wants the bigger fish, so they often move up in the chain to collect their bigger prize. The detective or arresting officer will often use the tactic of actually drawing up a warrant to show the initial arrestee, giving them the option of acting upon the warrant or having the person "cooperate" with more information that would push the investigation into the higher levels. Of course this "paper warrant" is not initially entered into the main database that the law enforcement officials use, mainly because they don't want anyone else to pick them up at a traffic violation or something else. In other words, it is often used when investigations are continuing. Whether it's ever acted upon or not is up to different circumstances. As we know, smaller drug offenses will often totally disappear with cooperation or "ratting someone else out," so the decision may sometimes be up to the person who has committed the offense, or it can be held off a while where the person may be allowed to turn themselves in.
3. One last example would be a burglary detective who has an arrestee or suspect whom they believe is involved in a chain of recent break ins around the area. They often show the arrestee a "paper warrant" (which could easily be enforced should they want it to be) for ONE of the break ins, in the hopes that the arrestee will cooperate and see them as a reasonable officer who is "cutting them a break" by holding off on the warrant so that they may either turn themselves in (which appears much better in setting bond) or even give them more information about other burglaries that others may have helped them with. The same example could be used for someone who passes 25 phony prescriptions around the area and eventually admits to ALL of their offenses in exchange for a "good word" from the investigator to the judge for a lower bond--etc, etc, etc. The lead detective would use the paper warrant for one of the charges in hopes of solving a bigger problem or getting more information about the specifics of these offenses.
The idea that there is no such thing as a "paper warrant" is not correct, although you'd be right in stating that it's a warrant that hasn't been acted upon yet or even sometimes made official. We have no idea how far the chain got along in the AH situation, so we're left to speculate with every single issue at this point in time. It's possible that the media was 100% flat out wrong, and it's also possible that someone heard about or leaked information that a warrant was in the initial stages and made the incorrect assumption that the next step would likely be taken.
Another possibility is that the police/DA's office was choosing to "use" the media involvement to put a scare into AH, which can often happen in cases of enormous media involvement. Information and mis-information is a powerful tool, especially if they wanted to apply a little more pressure. How effective that is remains a big debate, especially when someone like Michael Fee is your defense attorney.