How did Newton and Leibniz think about the integral? Did they only see it as an anti-derivative or did they also think of it as the area under a curve?
It was known even before Newton and Leibniz that areas under curves can be found by inverting the "computation of derivatives" (drawing tangents). In explicit geometric form this "fundamental theorem of calculus" was derived by Newton's teacher Barrow, see Barrow's Fundamental Theorem by Wagner.
Newton and Leibniz developed explicit symbolic methods for computing derivatives (isolated from the general process of drawing tangents), and anti-derivatives. To the extent that they conceptualized integral as such, it was identified with the anti-derivatives (what we call the indefinite integral), but, of course, it was used as a tool for computing areas, among other things. Euler still thought of it this way in 18th century. The idea of the definite integral does not appear until Cauchy in the 19th, see Kallio's History of the Definite Integral.
Newton proves the upper and lower sums converge when the function (actually curve) is positive and monotonic (decreasing) in Lemmas II, III in the Principia. Newton does not formulate the process in terms of integration per se, nor in terms of functions. It is a solely geometrical process. The proof is equally valid if the function is negative or increasing; and it is easily extend to the case where the function is piecewise monotonic (i.e., a finite number of monotonic pieces). Newton does not make these trivial extensions.
According to Kallio's description, which @Conifold references, Fermat's work resembles Newton's, although Fermat does not seem to bound the area with inscribed and circumscribed rectangles but just uses (something like a limit of) one sum of rectangles. Fermat also works algebraically, not geometrically in the Principia. Also it seems Newton did not develop further the limit of sums as integration; at least, according to Boyer, as cited by Kallio, future work was in terms of anti-derivatives as @Conifold says.