Is this a new problem or has it always acted this way? What kind of toilet (old or newer low water)? Are you on city sewer or septic, does your septic have a low vent at the leach field?
My experience with drain issues is that lots of information is needed to track the culprit. Sometimes to the point where a map is needed showing the positions of the fixtures and vent connections relative to each other, the diameters of the pipes and the routing of said pipes.
From your description, it sounds like the large volume of water from the toilet is filling an entire pipe (normal) which creates high pressure ahead of it and low pressure behind. The high pressure ahead, which sounds like your problem, should be able to escape via down stream vents and the low pressure does not appear to be an issue for you (more of a gurgling than a burping).
I tend to agree that this is likely a venting problem.
If you have been in the house for a while and this problem is only recently developed I would say you have a clogged vent line somewhere. I know you said you snaked the vent 25 feet, but a vent stack (depending on the vent diameter) can have multiple branches leading to multiple fixtures. I've also seen so vent lines that have shapes that would never work for water and that could never be snaked from above. Regardless if it is a clogged vent it is much more likely to be clogged near or at where the vent connects to the fixture drain than near the roof vent.
If this has always been a problem, I hate to say it, but it may be a poorly configured venting system.
In my current house most fixtures have a short run to the main drain line where they interface with the vent system. (if the runs to the trunk were longer or if figures shared a spur I might encounter gurgling.) So far this sounds a lot like your configuration. If most of your venting is through the main drain trunk or some other form of "wet venting" I wouldn't be surprised to learn that your layout looks something like this
Vent Toilet Sink
l l l
l l l
l_________l_________l__________to sewer
and the high pressure wave in front of the toilet flush is forcing air up the sink. Unfortunately, and AAV/ Studor won't fix this. A downstream vent may be needed.
I am a little stumped by the second and third flush not causing the same issue.
