I think it's all right. I know you gave an example so I am not sure if it was purposely left out, but I'd write the last name and level of training from people who you received and a report from and who you gave a report to.
"Received a report and paperwork from Smith, RN."
It's really not that hard to ask them for their last name or look at their name tag, and be honest "Can I have your last name so I can write down who I gave a report to?"
Like names, I think it's nice to include the department and room number (e.g. ER room 12) of the hospital too.
I used to write non emergent too, but the company I currently work for recommended that I did not write that. They prefer that I write "without red lights and sirens" or "without warning devices" or priority/CODE it is. Something about even if it's without red lights and sirens, it could still be considered an emergency, but one that doesn't require red lights and sirens. Since you are mentioning that you transported without red lights and sirens, you minus well mention how you responded too (even though to us, it's a given that you didn't respond with 'em).
In your first example, you forgot to mention the patient's position on the gurney.
I think it's standard to say the medication name, dose, route, who it was administered by, when, and how the patent tolerated it. I think it would be easier to give an actual time rather than saying one hour ago "1 mg Dilaudid by Smith, RN at 1400. The patient said it reduced her pain from a 10/10 to a 7/10". At least a name, dose, and time in case your patient begins to crash during transport and you have to transfer your patient to ALS or give a report in the ED. It's no fun saying "I don't know" or "It's in the paperwork" when you are giving a report in the ED or to ALS.
Do you have a section to write down your vital signs? Reading the PCR, it sounded like you only had one set of vital signs. It's nice to get the last one from the hospital (write a note like "VS prior to transport at sending facility by Smith, RN done at 1400 HR 72, BP 120/80, RR 16, Temperature 98.6F orally, SpO2 100% on Room Air", even vital signs you won't record that you can at least relay to the receiving facility (in my area, we don't regularly check temperature and EMT's are not allowed to do pulse oximetry so we don't have a section for it on our PCRs), a baseline set of your own on your own equipment, and at least one more to watch for trending. I believe medicare requires at least 2 sets of vital signs.
In my area, AOx3 would be considered confused. Anyhow, you could be specific to what they were alert and oriented to "alert and oriented to person, place, time, and event". Ah, I actually see you did it in the second narrative. Yeh, just like that... I liked how you mentioned incoherently. You could also mention if they talked fast/slow, soft/loud, and if they were clear/distinct. "Speech is slow and words are mumbled."
You said that the patient had an unsteady gait, but later on you said she can't stand. See a problem? If the patient can't stand, the patient can't gait :P. Is the patient unable to stand physically, is standing not recommended by the patient's doctor, does pain increase when the patient stands, did the patient refuse to stand?
Mention if there were any changes during transport.
Captain ToHellWithItAll is right, I don't see a physical exam in there. I don't see any physical assessment on the right leg at: Can the patient move it? Is there a scar from the incision? Does it hurt when the patient moves or stands on it? Is circulation, motor, and sensory intact? What is it like compared to the other leg? Any edema? Warm to touch?
I think we act like witnesses and should support/verify other documentation on the patient. In other disciplines of the medical field, your documentation represents how good of a provider you are. I think that some information is considered worthless if it's partial e.g. saying "RN" instead of "Smith, RN". I'm not suggesting you would go to court over a transport like that, but if you did, I am pretty sure the court would be interested in names rather than just level of training. It's difficult for me to explain what I think would be excessive for a PCR. Like I think if the patient told me about their 4 grandchildren during transport and how one of them is really good at dodgeball, I think that is worthless information to include on a PCR, although writing about the patient being able to talk clearly and coherently, how loud or soft they spoke, how fast or slow they spoke, etc is important (especially when transporting a patient with dementia). See what I mean? I think you are on the right track with your narratives.