Proxy - use another currency to hedge if cheaper, more liquid etc. Highly correlated with home currency.
Cross - trading one exposure for another. Could just be because you think one will appreciate more relative to home currency. You cross through an intermediate currency trade, but ultimately hedge the currency where you hold risk.
However in the curriculum for currency management (Economic analysis) it say “a cross hedge, also referred to a proxy hedge”. So it seems to me the curriculum says these two are the same?