Create anchor on "intersection" not working, is always off

Question

When I use the new anchor tool to create a new point on an intersection, it is never EXACTLY on the intersecting section, but off by a little bit. Sure, I could manually edit the anchor, but the time spent doing this many times over adds up and a slight alteration to the geometry occurs as well.
Both anchors were created on the same so called "intersection". I would appreciate it if someone could direct me as to how to make anchors on the true intersections.

Answer 1

Howdy.
Let me get this straight.
When you hover the Add Anchor Point Tool over an intersection, "intersection" appears beside it, and then you click, but the new anchor is not at the intersection. Hmm. If your preferences are set as recommended above, you do have a problem.
Your screen shot is pretty soft, and I'm not sure what is going on in your first example, but looking closely at the right image, this is what I see:
The arc coming from the left is an open path. You can see its endpoint. It doesn't intersect the straight path at all. The strokes are aligned, not the paths. When you hover over what you think is the intersection (there isn't one), the tool picks up this endpoint and snaps to it, placing the new point there. The distance between the straight path and the endpoint of the arc could be less than the snapping tolerance set in your Smart Guides preferences, so even if you hover precisely over the straight path, the new point could snap to the endpoint of the arc. This will depend on which way you are moving the mouse. If you move the cursor over the arc endpoint first, it will pick it up and indicate "anchor". It will do this until the cursor excedes the snapping tolerance, then it looks for a new target. So you cursor could be over the straight path, but still locked to the arc endpoint. And will place the new point there. Coming from the right, if the cursor passes the straight path first, and reads "path", even if you move it over the anchor point. And will place the point on the path. With two targets within the snapping tolerance, the unexpected can happen. If this guess is right, when you hover over what you think is the intersection, the cursor will indicate "path" or "anchor", but never "intersection". There isn't one.
Short story, strokes could be leading you astray. I usually do this kind of operation in paths only mode (Crl+Y). This way strokes and fills don't confuse the issue. The above image would look something like this.
I could be barking up the wrong tree, but I hope not. Otherwise, you could have a software issue.
Peace,
Lee