Why "rent out the entire rink?" That makes it into a formal event, makes you and the other parent the hosts, and yes, I can see how that could make the liability fall onto you somehow.
I would just issue an invitation for parents to bring their kids to skate at a certain time and say clearly that though other kids from the grade will be there, this is not a school-sponsored event and is not sponsored by your family or the other family, and everyone needs to stay with their own kids, pay their own way and sign whatever the rink requires. In other words -- it's not a party where you have the facility all to yourselves, it's just a matter of "If you want to skate at a time when friends will also be there, this is the time to come."
Ask the rink what the times are when they are least crowded. But I would not try to rent out the place out of concern that I would then be the official host and would have some liability personally.
When you say that "the rink does not have people sign waivers to skate," don't people at least sign something saying they accept that they skate at their own risk? I have never been to any kind of laser tag place, skating rink, or any other similar place that did not have you sign something like that.
If you want it to be "a we are all getting together at the skating rink" deal like you mention - make it just that, and don't rent out the place.