Quick ideas
Nicknames are personal, and not every short form has the same status. For Szilvia, the safest distinction is between Hungarian-style diminutives, spelling variants used as informal forms, and purely creative nicknames.
Hungarian-style short forms
Szilvi is the most intuitive short form because it simply trims the final vowel while preserving the Hungarian spelling. Szilvike follows a familiar Hungarian diminutive pattern and can feel affectionate.
International or simplified forms
Silvi removes the Hungarian sz spelling and may be easier in an English-speaking environment. Sylvie is actually an established French related name, so it works better as a cross-language relative than as a claim about Hungarian nickname tradition.
Creative family nicknames
Via and Vivi can work as modern family nicknames based on the sounds or letters in Szilvia. They are creative options, not established etymological forms.
Which nickname is best?
For Hungarian identity, Szilvi keeps the spelling most clearly. For an international setting, Silvi may be easiest. In real life, the best nickname is simply the one the person likes and uses.