Position oven rack in lower third and preheat to 325°F (163°C). Do not grease the 10-inch tube pan. Separate eggs, placing yolks in one bowl and whites in a clean stand mixer bowl.
Sift cake flour, 1 1/4 cups sugar, baking powder, and salt into a large bowl. Whisk in lemon zest until combined.
Make a well in the dry ingredients. Add oil, egg yolks, water, lemon juice, lemon extract, and vanilla. Whisk from center outward until smooth with no lumps, about 1-2 minutes.
Beat egg whites on medium speed until frothy. Add cream of tartar and increase speed to medium-high. Slowly sprinkle in remaining 1/4 cup sugar while beating until stiff, glossy peaks form.
Fold one-quarter of whipped whites into yolk batter to lighten. Gently fold in remaining whites in three additions using a J-shaped motion until no streaks remain. Do not overmix.
Pour batter into ungreased tube pan, filling two-thirds to three-quarters full. Run a knife through batter in figure-eight motion to remove air pockets. Bake 50-55 minutes until top springs back.
Immediately invert pan upside down over a bottle neck or funnel. Cool completely, at least 1.5-2 hours. Run a knife around edges and tube to release, then remove from pan.
Beat softened butter until pale and creamy. Gradually add powdered sugar, then lemon juice, zest, and salt. Beat until fluffy, adding cream as needed for spreadable consistency.
Place cooled cake on serving plate. Spread buttercream over top and sides using an offset spatula. Garnish with zest, lemon slices, or berries. Slice with a serrated knife.