I don't know the answer to this specific question with regard to the Orleanist wording, but it's worth remembering that there was a break in continuity between the (first) fall of the Bourbons and the rise of Louis Philippe: Napoleon Bonaparte. After all, Napoleon's official title was 'Emperor of the French,' not 'Emperor of France.'

Napoleon's choice to use 'of the French' was deliberate: an attempt to convey that he was not simply the sovereign of a geographic area by way of noble right or title, but rather that he was Emperor of the French people, ruling in their interest and by their consent. Napoleon gave the following oath as part of his coronation:

I swear to maintain the integrity of the territory of the Republic: to respect and to cause to be respected the laws of the Concordat and of freedom of worship, of political and civil liberty, of the irreversibility of the sale of the bien nationaux; to raise no taxes except by virtue of the law; to maintain the institution of the Légion d'Honneur; to govern only in the view of the interest, the wellbeing and the glory of the French people.

The wording of the oath puts him as both subordinate to the law and the French people. Now, the degree to which you see this as cynical political posturing will depend on your view of Napoleon as a person. What's certain, however, is that Napoleon was making a number of deliberate breaks with the Ancien Régime despite holding a coronation based on the Bourbon ceremony.

And we don't need to go far to demonstrate that the break with tradition was quite conscious. Lest we forget that the following year he took upon himself the title of 'King of Italy.' Again, the wording was deliberate. Napoleon was not sovereign of the Italian people; he was the sovereign of the state that governed Italy. This would go on to change, of course, with the French Revolution and Napoleonic wars playing a significant role in the birth of Italian nationalism, partially by way of the contention that perhaps Italy shouldn't be ruled by the Emperor of the French.

Regardless, hopefully someone can chime in with a more specific answer to your question, but I'd be surprised if the change under Louis Philippe wasn't a direct response to shifting attitudes on the relationship of the sovereign of France to the French people and a desire to emphasize a break with the previous regime, much as it was with Napoleon's own choice of title

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I was mostly drawing upon The First Total War by David A. Bell and Napoleon: A Life by Andrew Roberts, with my original train of thought going jumping back to a lecture by Professor Spang at Indiana University (sorry if you see this, I should have been better about actually attending class).

Edit: formatting, syntax