When a vampire starts its turn in direct sunlight (literally sunlight, nothing else), then that monster feature is triggered. Nothing else triggers it, because 4e rules do only and exactly what they say they do.

When it triggers, the vampire takes 10 damage. It so happens that this damage has the radiant type, but that doesn't do anything special by itself and doesn't have any effect on when the feature triggers. Because it has a type someone can take advantage of it (say, if an PC's attack adds a vulnerability to radiant damage to the target), but it does nothing by itself.

All the other things in your question have no connection to the sunlight-triggered damage effect, because they don't satisfy the literal meaning of the trigger. Only an ability that conjured literal sunlight and made it fall directly on the vampire's square would trigger the damage. A very bright light doesn't cut it, even if the power's descriptive fluff says it's as bright as the sun – it has to be actual sunlight, not just "the power and intensity" of the sun.

Lastly, radiant damage from other sources doesn't interact with the vampire's sunlight feature because radiant damage is not part of its trigger, only its effect.

As is noted in your answer, the mechanics here are pretty clear:

• The vampire takes normal damage from Sunbeam (6d8, with the option to repeat on subsequent turns) and Sunburst (12d6).

• In the case of Sunbeam, the vampire takes an additional 20 damage at the start of its turn, if within the 60-foot radius.

The vampire doesn't take extra damage from Sunburst, because there is no sunlight at the start of its turn. Simple enough.

Which is "better" depends on the situation. 12d6 will generally do quite a lot more damage than 6d8, but 20 damage / round narrows the gap almost immediately.

This does mean that in this particular instance, the level six spell may be a bit better suited to the situation than the level eight spell. Never fear, the eighth level spell will usually be a lot better (hits more targets for more damage, does other useful things).

Mechanics vs. Flavor

The mechanics above are pretty clear. But the mechanics may not fit the lore you have in your head about vampires (I think this is what SSD is getting at with his comment). As the DM, you have two options here.

1. Fit the lore to the mechanics

   The first is to fit the lore to the mechanics. Here, it's pretty simple... Vampires burn in the sunlight, but it takes a little sizzling before they start to hurt. Sunburst just doesn't shine the light on them long enough to cause problems (other than, you know, the massive amount of radiant damage and potential blindness).

   A longer dose of sunlight gradually sets them on fire, triggering their weakness.

2. Bend the mechanics to fit (your version) of the lore

   Sunlight hurts vampires, right? So go ahead and throw an extra 20 damage onto Sunburst. It's not the rules, but if it makes the fight more fun, then that can be fine.

Which you choose will depend on your group, and how often you bend things in their favor ("hey, 20 extra damage!"), vs. how often you bend things against them.