I'm not an expert but I'll give my 2 cents.
Having an idea that LEO = radiation safe, cislunar = radiation dangerous and VAB = radiation deadly is wrong. To assess radiation and it's effects you need to know what types of radiation you are encountering, what is each type's flux and energy distribution, duration and repetition of exposure and attenuation of each radiation type.
Van Allen belts traps charged particles and protects everything below it from charged particles. X-rays, gamma rays and neutrons are electrically neutral and are thus immune to VAB. They are attenuated by our atmosphere. That's why you get higher radiation dose while flying because densest part of atmosphere is below you.
Mercury had lowest dose because they were shortest missions with lowest orbits. Outside the protection of atmosphere (x-ray, gamma ray and neutrons) but well below VAB and under it's protection against charged particles.
Gemini had longer missions and higher orbits, thus larger dose. Orbital Apollo missions (7, 9 and ASTP are in this category)
Apollo and shuttle had similar durations (about 2 weeks). Apollo had to traverse Van Allen belts. You can and Apollo did go over radiation belts. Apollo 14 had higher dose because it went through the heart of VAB and had more active radiation environment because of recent solar event. Apollo 14 didn't fry because it crossed the belts with high speed reducing exposure duration. Also capsule made of steel, glass fibre and aluminum attenuated particle radiation from VABs.
Space starts at about 100 km up but atmosphere doesn't end there. It just gets thinner and thinner. Even quite good vacuum creates air resistance when you travel at speeds of few tens of thousands km/h. Because of that, satellites which need to stay in space for long, use higher orbits. Hubble had about 500 km orbit that slices South Atlantic Anomaly on every orbit. Shuttle missions to Hubble had still higher doses because each orbit skirted the SAA until they landed.
Space stations orbit in 400 km so they take hits from SAA too but mission durations are about 6 months to even a year so they have the highest dosages.
Comparing trips to Moon and Mars are incomparable because of different duration. Trip to Moon was about 2 weeks. In 60's they could predict solar activity for a week and used statistics to avoid major solar events during mission. Galactic Gamma Rays are powerful but rare so they don't cause much exposure during 2 week trip. Trip to Mars (one way) is about a year. In mission that long, major solar events are certain and doses from GGRs add up. That's why we could do 2 week Moon mission but 2+ years long mission to Mars needs improved radiation protection.
Lurky