Does Symbol require the target to be fully inside the sphere

If the Moonbeam spell required the actual moon to be shining down, and somehow focused its light (or similar), you would assume that its text would reflect that. No mention is made of this, nor of the restrictions that it would naturally apply to use of the spell.

This wouldn't just be a problem indoors - consider phases of the moon, daytime, cloudy nights, or even eclipses. In fact, it would be an extremely limited usage spell.

Contrast the Call Lightning spell (Thanks Tashio for providing the text):

PHB Pg.220 Call Lightning
A storm cloud appears in the shape of a cylinder that is 10 feet tall with a 60-foot radius, centered on a point you can see 100 feet directly above you. The spell fails if you can't see a point in the air where the storm cloud could appear (for example, if you are in a room that can't accommodate the cloud).

If you are outdoors in stormy condition when you cast this spell, the spell gives you control over the existing storm instead of creating a new one.

This specifically says that it creates a cloud, includes limitations requiring you to be outdoors or have an incredibly high ceiling, and includes an additional effect if there is a pre-existing storm. Based on this, it seems clear that Moonbeam has no particular source or restriction whatsoever.

Obviously, you are the DM, and you are free to add fluff or requirements as you choose. But I would urge you to consider how limited this spell would become before you do.

The sphere of fire from the flaming sphere spell is only 5 feet wide. So on a grid with 5-foot squares, it would affect a 3×3 area, with the sphere in the middle square. For ramming, the spell specifically says:

If you ram the sphere into a creature, that creature must make the saving throw against the sphere’s damage, and the sphere stops moving this turn.

This means you use your command of the sphere to move it directly into a creature's space (thereby hitting it) and they have to make a normal Dex save, and if the sphere hits the target before moving its full 30 feet you lose the rest of the move distance.