your first paragraph is completely wrong. the lens concentrates collimated light parallel to its axis at its focal point, regardless of where it falls on the lens. (and, strictly speaking, only at a single wavelength.) collimated light coming from near-axial directions gets focused more or less to a point on more or less the focal plane. but light at a single point doesn't have a direction, being a wave. there is in fact a very profound connection between the action of a lens and the 2d fft; see my sibling comment for more details
I don't think the idea that (idealized, camera) lenses focus light from distinct points in one plane (or at infinity) onto distinct points in another plane is 'completely wrong', but I'm open to being educated on my error.
A lens focuses light parallel to its axis onto its focal point; it focuses parallel light coming in off-axis to other points on the focal plane.
Alternatively, and equivalently, it focuses divergent light coming from common points on planes closer than infinity, onto matching points on other planes behind its focal plane.
Lenses bring parallel rays of light (alternatively, light from infinitely far away) to the focal point. They don’t bring idealized points to points.
One consequence is you can’t use lenses to bring anything to a temperature higher than the temperature of the source light. For example you can’t use lenses + moonlight to light things on fire.
your second paragraph is correct, and it is a special case of the convolution theorem; see https://en.wikipedia.org/wiki/Fourier_optics#The_2D_convolut...