I think this argument can never be won. "Layer 3 switch" is common terminology. But "switching" strictly speaking is a Layer 2 action. But sometimes we say that a switch is "switching packets at Layer 3" when it is doing a hardware action in response to IP layer information. We could go back and forth all day. So let's all be reasonable if possible.
I believe “L3 switch” and routers(L3-L7) are distinguished by architecture; L2 and L3 switches employ non-programmable “packet switching fabric” ASIC with CPU acting as a control system, while routers are generally a general purpose computer optionally with non-Turing-complete ASICs for faster packet processing.
Expectations of a “switch” is therefore that it’s not a dual core PowerPC box with 24-96 GbE ports on PCIe, running outdated Linux Kernel, and that it can’t do what such a bare metal box could do.
I'm not sure what is non-programmable about the packet switching fabric. It can be programmed to switch packets, which is what we want the device to do. It can also be programmed to route packets, which is done at the same rate as switching them (usually line rate). So we can call this "layer 3 switching" because it is the same process as the L2 switching but it is happening at L3. That's what a L3 switch is and does.
It can't do the same as a box with general purpose CPU, but it can do the thing you bought it for (routing) at line rate (hence the comparison to switching).
