Hello and welcome to BW forums
Your description sounds similar to the first 5 or so weeks with my DS. I had been trying to put him down for sleeps since day 1 really but it's normal for little babies to want to sleep on their parents so it is a long and gentle process to get there.
Whilst the PUPD (pick up put down) method is for older babies, as you said, there is a little confusion around it. From memory in the books Tracy calls shush/pat PUPD too because you do pick up when baby is crying and put down when calm. The difference with the full on PUPD method (for older babies) is that there is some time constraint in how long you hold and some adaptations for age, including not picking up but just putting down if baby sits up. With a young baby there is NO time constraint on how long you hold or how long you continue you literally hold for as long as needed, try to put down (and continue shush/pat in cot) and pick up again for as long as needed if baby cries.
In these early days it is about gently teaching LO she can sleep on a firm flat surface laying down rather than in arms or over your shoulder - getting her down at all is a step in the right direction.
It really does take time, shush/pat isn't a magic method which makes baby sleep, it is a method of reassurance and comfort which can be done in arms and continued in the cot so that she slowly slowly learns she is safe and you will always return when needed.
What my DP and I found possible was a very gentle movement towards having LO sleep on our lap in a horizontal position but relatively hands off (using shush/pat during the falling to sleep part) or on a blanket or pillow right next to our lap, touching, and keeping a still hand on (increasing firmness or starting shush/pat as needed to keep him asleep). Either of these involve staying with LO throughout the entire nap, they both involve increasing and reducing shush/pat through the nap as needed, but they both involve LO learning that mummy/daddy is right there, even if LO is sleeping flat by the side or on the lap rather than "in arms" or "over shoulder". It is a small step towards putting down in a cot.
I also found I could more often get LO to sleep and stay asleep if I put him on the floor by me on a folded blanket, staying with him as long as needed but it also meant I could nip off for brush my teeth or get a drink or bite to eat! You must not leave LO on a sofa or chair or pillow, but flat on the floor on a blanket should be okay and feels similar to the cot mattress so it is another step towards learning how it feels to sleep on that flat still surface.
There is a description here from a FAQ on regaining trust (I am not suggesting in any way you have lost trust, it's just a useful description. I would not use a pillow though for such a young baby, replace with blanket, stay with LO for safety):
3) Start with the first nap of the day. At the first sleepiness cue start your wind-down. Swaddle and Shh Pat your child and when they are calm instead of putting them in the crib put them on a pillow on your lap. Tracy suggests a firm standard size pillow. Sit this your back against a wall for support. Let your child know you are there and will be there to help them sleep. When they have fallen asleep (20 minutes atleast) uncross your legs and let the pillow plop slowly and carefully on to the floor. Stay sitting next to the pillow. For the next week atleast stay next to your child while she sleeps. This is a sacrifice that you make to regain your child's trust.
4) The second week do the same wind-down but put the pillow on the floor for your child to fall asleep on, not your lap. Again stay with your child while they sleep through naps and nights.
5) The third week do the same wind-down but put the pillow in the crib. When you lay your child down put your hand on them so they know you are still there. For 3 days stay by their side until they are in deep sleep. On the fourth day remove your hand while staying beside the crib while your child is sleeping. Three days later leave the room when your child has gone into deep sleep but if they cry for you go to them IMMEDIATELY.
6) The fourth week you should be able to put your child down in the crib without the pillow. If not, start again from the previous step.
from this FAQ
https://babywhispererforums.com/index.php?topic=126141.0If you do continue to put her in the cot (or when you move to cot after a period of sleeping on lap/floor by you) stay with hands on shush/patting for the whole nap if necessary. Gradually you can reduce the shush/pat.
You can begin by shush/patting in arms until fully asleep, put down and continue for a minimum of 20 mins (or full nap)
then move on to putting down drowsy but again shush/patting all the way to sleep and an additional 20 mins (or full nap)
at every stage you can pick up if baby is crying and start again in arms
Sometimes when baby is down they start to cry but are so close to sleep that if you put your hands on as though to pick up but barely lift their weight off the mattress they nod off, just put straight back down, continue to keep hands on either still and firm or patting for 20 mins.
You asked about shush/pat over the shoulder and when to move to cot - you can bring LO gently down from shoulder to horizontal sleeping in arms as a mid-step. The movement from should to cot is quite a big move and may disturb some LOs. Instead alter the way you wind down over time, shoulder patting, then horizontal patting then patting in the cot. I discovered with my DS very early on that he needed his "seven mile stare" to be with the same view point every single nap. He had fixed onto a spot behind my shoulder when I was sitting on the sofa so anywhere else he wouldn't relax...to change this I had to be persistent and do the wind down in a different place, standing right next to the cot so that his view point was consistent but I was close to the cot to get him in without a big change in position moving from sofa to cot etc. Not long after this I got him down before the seven mile stare so that his view from in the cot became the focal point - and he would stare at the same place every nap and if he woke mid-nap he stared again at the same spot and it helped him know where he was. I think Tracy said something about this in the book, babies needing to fall asleep where they will wake so that they are not suddenly disoriented by the change. It was true for my LO.
If she is sleeping longer than 2hrs then certainly make sure she is awake and feeding at/by 3hrs. She might sleep 2.5hr and wake naturally, I see no harm in this and do not see it confusing her day/night but I wouldn't go past a feed time by very long.
I hope this helps to answer some of your questions and clear up a bit of the confusion.
It is a very gradual process and no quick fixes but you'll get there.
