Doffy doesn’t care for being a pirate. His only aim was to tear around the world around him. Hence him arming Kaido to make the greatest war in the world.
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Doflamingo isn’t interested in being a pirate, he becomes what is expedient to achieve his aim of tearing down the celestial dragons.
Once again does not negate, dude, these statements have a context.
Doflamingo´s ultimate goal is to pull down the Celestial Dragons, he does not care the way he does it.
He still chose the way of pirate because it´s the most logical and viable thing to do.
Pirate King is not his ultimate goal, he cares about it in regards to getting the power to establish his actual goal.
In contrast, Bellamy is a new age pirate through and through and only wants to advance within that mindset, hence the difference.
Characters have a context.
And the war stuff with Kaidou is inconclusive, it was never implied once Doflamingo´s interest was beyond a business partner for Kaidou, that´s just your one-sided interpretation.