Here is further evidence that spells like Moonbeam, Cloud of Daggers, Web, Sleet Storm, Spirit Guardians, etc. should not deal damage at cast time:

The spells that do deal damage at cast time state so explicitly, such as Insect Plague:

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When the area appears, each creature in it must make a Constitution saving throw. A creature takes 4d10 piercing damage on a failed save, or half as much damage on a successful one. A creature must also make this saving throw when it enters the spell's area for the first time on a turn or ends its turn there.

Other examples include Grease and Incendiary Cloud.

The wording for the delayed effect is utterly consistent across both categories:

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enters the spell’s area for the first time on a turn or starts its turn there

The wording for the explicit on-cast effect, only present in the latter category, is also utterly consistent between spells:

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when [...] appears

With this level of consistency in the wording of the timings, it is clear that "enters the spell's area" does not imply the moment of casting, just as Jeremy Crawford confirmed.