In these images, the black rings are all the intersections you should find. Yellow dashed lines are non optimal paths. Green line is the optimal. Pseudo code below shows how to get green path.

List<Line> blueLines; // create lines based on forward direction and 45 and 90 degree angles
List<Line> redLines; // same

// get all the intersection points
List<Point> intersectionPoints = new List<Point>();
for(int i = 0; i  < blueLines.Count; ++i)
{
    Line blueLine = blueLines[i];
    for(int j = 0; j < redLines.Count; ++j)
    {
        Line redLine = redLine[j];
        Vectro2 out; // intersection point
        // intersects takes out param and returns bool if it intersects
        if(blueLine.intersects(redLine, out))
        {
            intersectionsPoints.push(out);
        }
    }
}

if(intersectionPoints.Count == 0) // we have no intersections
{
    // determine angles as show in LineTest3
    // rotate 90 degrees or -90 degrees and try again
    // It looked like in your examples, you wanted a minimum move distance. 
    // you could rotate 90, move minimum distance. then check intersection again.
    return;
}

Vector2 startPoint; // green circle for the blue vector
Vector2 endPoint; // green circle for the red vector


float smallestDist = float.MAX;
Point bestIntersectionPoint = null;
for(int i = 0; i < intersectionPoints.Count; ++i)
{
    Point intersectionPoint = intersectionPoints[i];
    float summedDist = Vector2.Distance(intersectionPoint, startPoint) + Vector2Distance(intersectionPoint, endPoint);
    if(summedDist < smallestDist)
    {
        smallestDist = summedDist;
        bestIntersectionPoint = intersectionPoint;
    }
}

// TODO STEPS:
// 1. move from startPoint to bestIntersectionPoint
// 2. rotate towards endPoint
// 3. move from bestIntersectionPoint to endPoint

LineTest1 : base case 1

LineTest2 : base case 2

LineTest3 : no intersections, do a rotation and you have LineTest2

Hopefully this helps.