I had the injection (despite my birth plan saying I didn't want it) with DD1, it was a classic over managed ventouse delivery, baby yanked out, cord clamped and cut instantly and simultaneous injection and cord traction. My placenta was allegedly intact, but I did have to have some "clots" manually removed (the Dr just shoved her hands inside me and rummaged about ) and I did get an infection (inside, not in my stitches) which needed antibiotics to clear. I also bled out aprox 850 mils (which is a serious bloodloss) so had to stay on an synto drip for what felt like ages after DD1 was born. I bled for a good 6-8 weeks.
I had a homebirth with a physiological third stage with DD2. The cord pulsed for about 15 minutes and once it stopped we clamped and cut it and I had half a dozen (if I'm honest, very painful) contractions and it slooshed out without me actually having to actively push, but I was sitting upright (cross legged!) rather than lying down like you tend to be in hospital. My bleeding was much lighter, estimated bloodloss a birth was about 200 mils, and long term I only bled for about 3-4 weeks. No infection that time either.
Michel Odent is not in favour of the injection. In Birth Reborn he does go into detail about how the injection can interfere with the release of natural oxytocin, and how the synthetic oxytocin only prompts the uterus to contract, whilst the natural oxytocin does more (I can't remember it exactly and I can't find my copy of the book to quote it!).
I have a (totally unfounded, unproven) theory that the research that states that the injection reduces bleeding was done in the 70's when virtually no-one breastfed and babies were whisked away to nurseries within minutes of being born. With no skin to skin and breast/nipple stimulation the uterus wouldn't be prompted to contract and this would lead to bleeding. So we started giving synthetic oxytocin to replace that which isn't being released due to the lack of stimulous.
If we do it "natures way" and leave the cord alone (I also wonder if allowing the blood to transfer to the baby "empties" the placenta making it detach easier) until it stops pulsating and put the baby to the breast as soon as possible after birth that the placenta detaches and the blood vessels in the uterus clamp and this reduces bleeding to a minimum. After all, why would we evolve a birthing/placenta expelling system that is likely to make us bleed to death? It's far more likely that if we do it the way it is designed (by evolution/nature) to be done that 9 times out of 10 it will work just fine.
So, for me, go for physiological third stage, including leaving the cord to stop pulsating before clamping and cutting it, with the injection on standby just in case things don't move on thier own!