Avoiding terms like proper and correct, cos its all good just a matter of what we do with it ...
... you will probably find you are best off with your right arm resting on the guitar about where the bridge would extend to if it were long enough, bring your fingers to the strings about where they go over the rosette (the pattern round the soundhole) and give your wrist a slight lift so the fingers can work at each joint. Then aim each finger stroke in towards your elbow, roughly 45 degrees to the string. If you experiment with dropping the hand as you did before you should find both the tone goes thin, the wrist gets stressed, and the stroke is less stuck on the string and more of a glide. In other words there is a similarity of interest in the better (normal) tone, the most relaxation and the easiest stroke. This is especially the case with nails.
Everything in this game takes getting used to, its one of the things that makes it a 'classical' instrument, where we tend to use consciously developed techniques, rather than a folk instrument, where one tends to go with natural tendencies.
Simon Ambridge Series 40 (2005)
Trevor Semple Series 88 (1992)
Louis Panormo (1838)
Alexander Batov Baroque Guitar (2013)
Simon Ambridge 'Hauser' (2018)