Minkowski diagram and length contraction

What you've missed is that the distance along the $x'$ axis is not the same as the distance along the $x$ axis. The locus of events that are 1 unit of proper distance from the origin is a hyperbola. This can be used to calibrate the $x'$ axis. See calibration hyperbola.

One picture is worth a 1000 words here...

The important point is that we make a snapshot of the moving object in a time coordinate which is not its proper time. Indeed, if we drew just one image of the moving object in a cut given by $t' = \rm const.$, we would get a projection on the $x$ axis longer than $l_0$. But we must use $t = \rm const.$ instead if the measurement is done in the un-primed coordinate system. Thus, we measure the "front" of the object earlier (in terms of its proper time) than the "rear". The difference in $t'$ allows the latter to cover some extra distance, putting the two events closer in $x$. If the math is worked out, it indeed gives $l_1 < l_0$.