If it helps to clarify?

The source of wealth is only one part of the culture surrounding the "old money" mentality (and as TiceNryOSIS has pointed out, 2nd generation absolutely wouldn't be considered "old" by these standards). Think of it as the main gates at a housing complex - in order to get into a home, you certainly have to get through the front gate (have the financial means), but you also need to actually unlock the door...

There are general values, unspoken etiquette, and a lot of generations-old relationships between families that contribute to the culture often colloquially referred to as "old money".

Arguably, Trump doesn't hold these values, understand this etiquette, or maintain these types of relationships, and he never has. He's generally looked down upon among those who do.

There is one theory that views the majority of Trump's actions throughout his life as an attempt to "finally" gain access to this world - which he may see as legitimizing - up to and including his political aspirations.

Source: Family and some friends in "that world"