Michael - thanks for your question. I am going to move it to a more appropriate product-specific community where it will receive better visibility. And apologies for not having seen this sooner. I have discovered my alerts for postings in this community was unknowingly disabled.
I have done this three ways.
1) The easiest is to put a small keep out on the pins you don't want to connect to the plane. I use circle just smaller than the pad. Downfall is if component gets moved later, you have pinhole in plane.
2) Edit pad stack on component to have annular ring with no spokes on the layer you do not want a connection to.
3) Edit pad stack to have oval pads with same height and width on layers you do not want connection to. Go to plane setup options and say do not connect to oval pads.
First, are your plane layers defined as split/mixed planes? If you want both agnd and gnd nets on the plane it needs to be defined as a mixed plane.
"My goal is to have layer 2 be a plane with agnd and gnd and only connect to agnd at those 2 points." - It sounds like layer 2 has a different net defined on it then agnd where the connections are not being made.
You need to define a spécial PADS STACK for the pin you want to connect/not connect on a spécific layer.
- edit the part decal:
1-ADD the pin concernedin the pin list (B13 in the example)
2-ADD the layer concerned(Layer 3 in the example)
3-SPOKE = 0
4-SPOKE WITH = any positive value. ex:1
As a result you should see ,as images show,pin B13 connected to plane P5V (magenta) on inner layer 2 beside GND (green)plane on the same level.
On the second example you should see pin B13 NOT-connected to an new plane P5V présent on Layer 3 beside P15V (bleu)
I'do not have problems with that configuration in may jobs.
Have a good day/Bonne jounée.