Then we can just pick a new place for that center point, and transform that back into a non-overlapping position for the original rectangle. This lets us resolve the collision in one pass, touching the x coordinate only once and the y coordinate only once.

if(first.Bounds.IntersectsWith(second.Bounds) == false) return;

// Compute the center of each rectangle:
var firstCenter = new PointF( first.Left  + first.Width  /2f,
                              first.Top   + first.Height /2f );

var secondCenter = new PointF(second.Left + second.Width /2f,
                              second.Top  + second.Height/2f);

// Compute the offset from the second's center to the first.
var offset = firstCenter - secondCenter;

// Clamp this offset to the closest point on the enlarged second
// rectangle's perimeter. First horizontally...
if (Math.Abs(offset.x) < (first.Width + second.Width)/2f)
    offset.x = Math.Sign(offset.x) * (first.Width + second.Width)/2f;

// ...then vertically.
if (Math.Abs(offset.y) < (first.Height + second.Height)/2f)
    offset.y = Math.Sign(offset.y) * (first.Height + second.Height)/2f);

// Now form the top-left corner of the first rectangle at
// this new position.
var corner = new PointF(secondCenter.x + offset.x - first.Width /2f,
                        secondCenter.y + offset.y - first.Height/2f);

first.SetPosition(corner);

Then we can just pick a new place for that center point, and transform that back into a non-overlapping position for the original rectangle. This lets us resolve the collision in one pass, touching only x or y, and only once - rather than stacking a vertical collision resolution after a horizontal resolution using outdated info.

We call this the "minimum translation vector" - using the smallest adjustment possible to remove the overlap, whether that's horizontal or vertical.

if(first.Bounds.IntersectsWith(second.Bounds) == false) return;

// Compute the center of each rectangle:
var firstCenter = new PointF( first.Left  + first.Width  /2f,
                              first.Top   + first.Height /2f );

var secondCenter = new PointF(second.Left + second.Width /2f,
                              second.Top  + second.Height/2f);

// Compute the offset from the second's center to the first.
var offset = firstCenter - secondCenter;

// Compute how deeply we're overlapping, horizontally and vertically.
var penetration = new PointF((first.Width  + second.Width )/2f,
                             (first.Height - second.Height)/2f);
                - new PointF(Math.Abs(offset.x), Math.Abs(offset.y));

if (   penetration.y <= 0 
   || (penetration.x  > 0 && penetration.x < penetration.y)) {
    // If we're only overlapping on the x,
    // or we're overlapping on the x less than on the y,
    // then correct our x position for the minimum safe correction.
    offset.x = Math.Sign(offset.x) * (first.Width  + second.Width )/2f;
} else {
    // Otherwise, our minimum safe correction is vertical.
    offset.y = Math.Sign(offset.y) * (first.Height + second.Height)/2f;
}

// Now form the top-left corner of the first rectangle at
// this new position.
var corner = new PointF(secondCenter.x + offset.x - first.Width /2f,
                        secondCenter.y + offset.y - first.Height/2f);

first.SetPosition(corner);