Does the statement of the person giving the zakat, "Distribute this money as sadaqah" to the one entrusted with paying out the zakat, change the intention of zakat to sadaqah, thereby obligating him to pay out the zakat again?

There is no contradiction between intending Zakat (definition) and saying "sadaqah (definition)," for Zakat is an obligatory sadaqah (charity). What matters is the intention of the heart, not the utterance of the tongue. So, if the giver intends Zakat in their heart, it is sufficient, even if they articulate something else. The intention, by consensus, resides in the heart, and what the tongue mistakenly utters does not affect the validity of the intention.