No. When you lerp, the interpolated point lies on a straight line between the two endpoints. If the endpoints are on the surface of the unit sphere, the interpolation will trace out a chord of the sphere, not an arc.

You can renormalize after lerping, and this is good enough for many cases (e.g. shaders). It doesn’t however give you a linear relationship between the interpolation parameter (t) and the change in angle (or equivalently, arc length on the sphere) - the nonlinearity being greater the wider is the angle between the endpoints. You can counteract this by using slerping at the cost of some additional math.

Either method (renormalize or slerp) also blows up if you try to interpolate between antipodal points, such as (1, 0, 0) and (-1, 0, 0), as there are then an infinity of possible shortest paths. If you need to handle such points you can special-case it and choose one of the paths arbitrarily.

The question pretty much sums it up, is it true that:

for l > 0 and 0 <= t <= l. I can work out the maths, but I got some weird results, so I think I have an error somewhere along the lines. I’ve googled, of course, and found a book saying one should renormalize, but does not state why. If the above does not hold indeed, I would ask you to provide some hints as to what kind of interpolation, if any, will yield this result.