Do I need a new cassette?

[HLaB]

What does skipping mean in this context?

Its when a chain doesn't fit exactly into a cassette due to one being worn. You often dont notice it until the bike is under load when the chain starts slipping or skipping over gears in the cassette.

 

[RumbaAzul]

He pointed out the silver ridges as being wear and tear

Trust your own research on this one.
He's wrong, you're right.
How you get this through to him without pissing him off is another matter entirely.
But? losing him as a mechanic would be no great loss. You know how to research stuff at least.