Your confusion comes from the fact that there are vectors and there are points.
Unfortunately, THREE doesn’t have a class Point3, it uses Vector3 for both, the docs occasionally saying that something is a “a vector representing a position” (there is no such thing, strictly speaking).
A point represents a position in space, it has x,y,z coordinates for that (in Cartesian coordinate system). “Look at” is a point.
A vector represents a direction and a length (along that direction). A vector can be described by either two points (the start and the end), or by one point (the end), in this case the start point is assumed to be (0,0,0).
In THREE, Vector3(x, y, z) can either be a point (to look at, for example) or a vector that goes from (0,0,0) to (x,y,z).
The camera’s initial z-vector is (0,0,-1), starting from (0,0,0), going to (0,0,-1).
When you apply the quaternion to it, it rotates the vector along the camera’s current line of sight. This line of code should always be:
const forward = new THREE.Vector3(0, 0, -1).applyQuaternion(camera.quaternion);
you can not put any other coordinates in this code, the quaternion should always rotate the original forward vector which is (0,0,-1).
If your camera is not at the origin, (you moved it), you need to move this vector as well, to re-attach it to the camera, if you like. The start point of this vector will be (camera.position), the end point (camera.position + forward).
All look at points will lie on this re-attached forward vector, for example you can choose (camera.position + forward) point to be the new look at point.
Originally, all look at points lie on the negative z-axis, you can not find them anywhere, they all lie on (forward) vector (0,0,-1).
When you call camera.lookAt(), all it does (-ish), it changes the camera quaternion. So after that, you need to update (forward) vector using the code above and then re-attach it to the camera, then everything will be in sync.
If it’s too much, you probably could find some video to explain how to work with vectors first, that will help to understand how to program them.