"High finance" comes from the French "haute finance". Finance is a multifarious industry, with many layers servicing the entire economy. "High finance" is most easily contrasted to retail finance, for example retail banking where the bank serves individual customers or businesses, taking deposits and making loans.
Traditionally "haute finance" involved lending money directly to the king or country, and the financial transactions between countries. Finance is much more than just lending money nowadays and has many other large players than just countries too, so the meaning of high finance has also evolved.
You can loosely define high finance today as financial services between large institutions (instead of individuals), and involving large transaction sizes. So within finance, one hears terms like "institutional investor" or "institutional clients". The quintessential institutional client in Singapore's context is
GIC or Temasek, but it will also mean hedge funds. Given that these institutions can trade/transact in large sizes and consistently, they will get the best pricing in deals.
Private banking is typically not "high finance". Individual clients are rarely considered as "institutions" unless you are Warren Buffett or a few such individuals. There are also important regulatory and compliance reasons to differentiate between individuals and incorporated institutions. Even most billionaires would need to create a family office to become an "institutional client".
Within banking, I would say that if your client is
GIC or
UOB Asset Management, rather than Mr. Tan or Garden Restaurant Ltd, then you are operating in "high finance". But frankly, this term is rather ostentatious and rarely used by bankers. We just say "institutional clients".