Gâteau basque and Basque cheesecake are different cakes. The first rather comes from the French part and the second rather comes from the Spanish part (Basque: euskal gazta-tarta, Spanish: tarta de queso vasca).

Basque cheesecake

Gâteau basque (usually with cream filling, sometimes with local black cherry jam)

If anyone goes to the French Basques country, I highly recommend buying the ones from Maison Gastellou, both the cream version and the cherry version are marvelous.