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In the Schengen area, the relevant distinction is between flights to non-Schengen destinations and flights to Schengen destinations. If you have two stops in the area, your journey includes a flight between Schengen destinations and you will need to go through the “external border” checkpoint to catch that flight. There will be no passport check at the second airport and once you have crossed the border at the first airport, there is nothing stopping you from leaving, which is why transit privileges do not apply.
Whether the next destination is in the same country or in another Schengen country does not make a difference and your UK visa and all the other rules and regulations regarding transit are not relevant in this situation either. As an Indian citizen, it means that you do need a type C Schengen visa (and not merely an “airport transit visa”).
See also Do I need a visa to transit (or layover) in the Schengen area? for a full explanation of the rules.