Can I actually have my baby in the birthing pool?
This is possible but there are a few factors that you will have to bear in mind. The midwife who is assisting you should be experienced in this and comfortable with the idea. There are some midwives who are happy to help look after you through labour in the pool but are not comfortable with conducting the actual delivery under water. That has to be established beforehand.
Moreover, there are some hospital units which have a policy of not allowing underwater births. The practice in those hospitals is therefore to get out of the pool for the second stage. Again, this has to be established well in advance. Of course, if this does not match with your expectations, you can seek an alternative midwife or hospital.
Theoretically, birth under water is safe as long as all the conditions are met. The baby will not breathe under water (dive reflex) and therefore the risk of aspirating water into the lungs is not there.
The temperature has to be strictly maintained and the baby allowed to surface immediately after birth. Prolonged immersion may allow the placenta to detach, which will cut off the oxygen supply to the baby, provoking a gasping reflex and therefore this is never allowed to happen.
When appropriately conducted by an experienced midwife, this procedure should be safe. However, because these strict conditions may sometimes not be met, some units are cautious in allowing it.
There is also the fact that you may yourself prefer to leave the pool for the second stage, which is fine. Keep an open mind in all these things. If your instincts when the time comes are that you would rather have your feet firmly on the ground, by all means say so and the necessary assistance will be there.