It is not a legal open write up, hence real no legality issue. Flight Attendants at my place are not trained to write up items in the Maintenance Log, like at your place I am assuming, or can they legally sign for it. Proper format doesn't matter.
I would ask her if she did write something down and if she did I would write her up and make sure she had disciplinary actions taken against her. I would demand she was sent back for retraining actually on CRM.
I would write an ASAP report just in case, but wouldn't lose sleep over it.
Unless you're changing pages every leg, I don't see how you would be able to tell that it was written up on the second leg. And, since you are able to fit 5 legs on a page, you'd notice it pretty quick. Unless the FA was smart enough to start a new page once you got the write up on the first page..... hmmmm.... but then you'd notice that you're on legs 3-5 on a new page, and you don't remove the carbon copies yourself.
I see this as one question that generates a couple issues.
First, let's talk about logbooks. The FAA mandates that a company keeps a logbook on each aircraft. Like personal logbooks, they outline what records need maintained at a minimum, and it's up to the individual company how to format, record and add any extraneous data they want to keep. Naturally, all this information is laid out in the Operating Manual for the airline (Trivia: Did you know that there is only ONE operating Manual for an Airline? It is broken down into various volumes to give each workgroup what they need, but the various volumes are viewed as one manual by the FAA.).
Some airlines, like the one I came from previously, had a log page that recorded employee numbers, flight times, trend info, and discrepancy info for multiple flights throughout the day. The airline I'm at now records the same information, but there is only 1 flight per logpage.
Other airlines, and I'm going to guess PSA is like this according to what BobDDuck wrote, have purely a maintenance book, to include employee numbers, but no flight times as the FAA approves the station or ACARS times in the airline's computer network as approved record keeping.
In the third case, I can see how Bob (or whoever told him the story) got blindsided by flying a couple legs with an open write up.
The other factor is the "legality" of the write up brought up by our resident wookie.
Each company's Operating Manual designates who may enter a discrepancy in the log (CA, MX, "crewmember" or "flight crewmemeber"). The previous airline either the CA or MX were the only ones allowed to enter discrepancies (the F/Os might do it under the CA's discretion, and then have the CA sign the block - IMO a great technique for helping develop our Captains in Training). This airline, the CA, FE or MX will make an entry into the discrepancy block, and just about everyone does.
The company operation manual will also designate if a flight may occur with an item in the "discrepancy" block, or if it MUST be closed out prior to continued flight. I've only worked for 2 air carriers, and at both, all discrepancies must have a corrective action. That corrective action might be an inspection, MEL, DMI, CDL, or repair, or another approved method to veirfy that the discrepancy has been dealt with appropriately.
From an FAA standpoint, and especially after the maintenance records issue, I doubt they'll really care who wrote the discrepancy in the book, and if you argue, they'll wonder why the FA or whoever had to writeup something the CA refused to (Argue this point all you want, but I guarantee the Feds will come in from this angle, and you'll lose...). All the FEDS want to do is to verify there is a corrective action for each discrepancy so they don't have to do more paperwork than the minimum to keep their jobs.
The ASAP will keep the FEDS at bay in the case the Company self-discloses or the open writeup is found during a records inspection (which under the new world order of commercial aviation this percentage of likliness is increasing exponentially).
The ASAP report will provide several things: 1) It will alert the company and the FAA to an obvious issue in training individuals. 2) It will provide protection to the flight crew from punititive action. 3) The ASAP news letter will alert other pilots to this incident, allowing the collective group to learn as a whole. 4) The accepted ASAP report (which under the ASAP acceptance criteria, this will definately be accepted) will convert to a NASA form, getting this info into the ASRS system. This allows trend monitoring of related issues inside the company's safety system, as well as the ASRS system. Also, the ASAP program can recommend changes to current company procedures. Perhaps a "Cabin Discrepancy" form might be created out of this. It happened at my old company. We gave the F/As a pad that they could write up cabin issues and slip it to us inflight. Then we could notify MX and have the issue resoved at the inbound station.
While ASAP is a union negotiated benefit, ASRS (NASA form) is a benefit to everyone.
On a side note, reporting an individual who did this for punitive action furthers the cockpit/cabin (or CA/FO or Pilot/MX) divides. Perhaps a report of the incident with a recommendation purely for training, and taking the tack of an honest mistake from an aviation neophyte (most FAs) will collectively strengthen the knowledge base the group, and foster working together versus raising a wall to CRM.