Spacetime interval is independent of Lorentz transformation. Spacetime interval is an invariant property of a spacetime path (when speaking of interval between two events in SR, implicitly you imagine a geodesic path between them). It is independent of coordinate system, reference frame, or observer. Lorentz transform is the particular form coordinate transformation between different inertial frames with standard coordinates, when there is also no translation or rotation.
The last sentence of your comment quoted above shows the confusion of separating a formula from its meaning. Length contraction and time dilation are very different things, and both are different from what I was referring to, even though they happen to have the same formula in inertial frames. Length contraction refers to disagreement between frames on measured distance between two objects (or ends of one object), each frame needing to do a series of operations to accomplish such a measurement, using their own, conflicting, simultaneity determination. Time dilation refers to the ratio of a moving clock rate to time rate in some inertial reference frame (sticking to the simple SR case). Time dilation for the same clock will differ between different inertial frames. What I was referring to was computing the proper time along a specific clock path, using some inertial frame. This will come out the same in every inertial frame (in one frame, the dilation may be greater on average, but then the coordinate time will be greater, so the integrated total will be the same; that's what it means to say proper time = time measured on some specific clock between two identifiable events in its history, is invariant. )
When I threw out the idea of trying to call the zero interval along a null path its proper time, it was correctly pointed out that I was proposing there was some type of clock that could follow along a light path - that's what proper time means
. I accepted that my suggestion wasn't meaningful.