If you've had a normal, straightforward natural birth (no induction, no assisted delivery, so a spontaneous birth with little to no intervention), it usually considered safest to have a natural physiological 3rd stage (no injection to help you deliver the placenta). Usually in this case, your body is doing fine producing its own oxytocin and there's no reason to intervene on what's working, except in cases where a PPH might be suspected (like if you've had a previous one). If you've had various sorts of intervention, including induction, it's often considered safer to have a managed 3rd stage with the injection. Induction alone increases your risk of PPH by 80% because all the synthetic hormones can lead to uterine atony, which is basically where your uterus gets too tired to contract down and expel the placenta because the synthetic hormones cause it to contract in a way it never would naturally. So often in those situations, it's considered best to intervene to head off any complications that could be caused by the intervention itself or just through the complications of a difficult labour.
With my first, I opted for a natural 3rd stage (no injection), but after about an hour, the placenta still wasn't coming and they were concerned about blood loss and the risk of a retained placenta, so I got the injection and it came about 25 minutes later. I had a home birth, so all perfectly natural and straightforward, and no complications or interventions, but my daughter was back to back, so I had a long drawn out pushing stage while she was turning. Everything was fine, but it just meant my uterus was more tired and worn out than it needed to be for the placenta to come on its own. We waited to see if it would, but when it didn't and I was losing some blood, I got the injection and it was fine. I lost about 550ml of blood, which is technically a small PPH, but not a serious one. I was fine. It was never an emergency situation of any sort. The injection sorted it out and all was well. I plan to have another home birth next time and likely won't opt to have the injection right away, as for me, I think it makes sense in the context of an otherwise easy, natural birth, no to then intervene at the end unless there is a reason to. But you can always decide in the moment and get it. Usually if you decide in advance, they'll cut the cord very quickly and give you the injection. That's important if you genuinely need it, but it does mean you lose the chance for optimal cord clamping, so your baby loses blood and is at more risk of anemia. But you can always request to wait a few minutes, have some skin to skin, let all the blood drain back from the placenta to baby and see if it's coming and how you feel, and then if you need it, cut the cord and get the injection then. I waited an hour and all was fine, so there isn't a rush if you want to wait and see, but at the same time, yes, it's a very standard thing to do, especially in situations where there has already been intervention (like with induction) or where there were other complications, and in those cases, it can be the safest approach. It's just a matter of balancing what you feel your body needs in the moment, your wishes for optimal cord clamping and some hands off time after birth, and just what you feel comfortable with. It may be you find it reassuring to know you can minimise the risk of another PPH and that will make the whole experience more pleasant without it weighing on your mind while you're giving birth.