The short answer is – there aren’t.
Many corners of the internet encourage people to continue lying on their back, perhaps with a pillow under their bum after intercourse. The speculation is that this allows more sperm access to the womb and increases the chances of pregnancy.
Does it work? I don’t know.
But, if it does, you would only need to lay in this position for about ten minutes. After that, all the “best” sperm will be gone. The vagina, after all, is a very acidic place and sperm do not last very long. After ten minutes, the remaining ejaculate will naturally come out when you stand up. But all the good sperm is gone.
Intrauterine insemination is different: The sperm have already been placed up into the womb. No amount of further lying on your back will change that.
Should you have a quiet day that day? You can! But only because you would like to have a quiet day.
After all, hopefully the sperm will be meeting the egg that same afternoon. But it takes a full five to seven days for the fertilized egg to float down and implant.
In other words, you are not going to achieve implantation that day, so there is no particular reason from a fertility perspective that you could not have as active a day as you had in the days leading up to that procedure.
Intrauterine inseminations do not have the highest success rates in a fertility clinic. IVF generally does.
IUI is much more straightforward and there are huge advantages. It’s a less expensive procedure. It’s less invasive, and it interrupts your life less. If having a quiet day the day of the insemination appeals to you, please take it! But it’s not a requirement for pregnancy.
The appropriate choice is to take on whatever level of activity emotionally fits the best choice for you.