TL;DR If you are driving trains, operation railway infrastructure or are a person interested in signalling and train protection, you can determine the two types by looking at the details.
Automatically controlled level crossings are activated by the train passing axel counters or other contacts (on branch lines, level crossings after halts are sometimes enabled manually by a key switch). In Germany, automatically controlled level crossings either have signals for the train driver to indicated whether the activation was successful, or a sign indicates the location where the activation happens (+ reporting to the dispatcher if the level crossing is defunct). Very likely, other countries have such signals as well.
Remotely controlled level crossings are fully integerated in a signal box, i.e. the dispatcher activates the level crossing before the signals display a passing permit.
In Germany, automatically controlled level crossings should be closed for less than 3 1/2 minutes at most and have barriers blocking half of the width of the road only. After that time, the number of car drivers passing closed half-barriers rises. Remotely controlled level crossings can be closed longer (almost always signals depend on them, the barriers must be lowered before the train passes the distant signal). They are usually closed for four minutes and more, and have barriers over the complete width of the road.