Is a vow to pray twenty Rak'ahs upon overeating, with the addition of "seeking nearness to You," considered a vow of devotion or a vow of obstinacy?

Vows of piety are of two types: First, the vow of recompense, which is the commitment to an act of worship in exchange for the occurrence of a blessing or the warding off of a misfortune. Second, the commitment to an act of worship without it being contingent on anything. Both types must be fulfilled.

As for the vow of lajāj (obstinacy), it is the contingent commitment to an act of worship by doing or refraining from something, either to prevent oneself from an action or to encourage oneself to it. In this type, the one who made the vow has the choice between fulfilling what they vowed or offering an expiation for an oath (kaffārat yamīn).

The vow mentioned in the question is a vow of lajāj, and in it, one has the choice between fulfilling it or offering the expiation for an oath. Fulfilling it is more preferable. It is worth noting that the phrase "seeking closeness to You" does not affect the ruling.