Displaying names inside pads in Altium
I don't believe it's possible to do what you want, because a schematic symbol is not a PCB footprint.  As you know, the schematic has "pin numbers" and "pin names", whereas footprints just have the "pad designator" (typically "1", "2", "3", etc, but can be anything).  You assign one of several PCB footprints to a schematic symbol (e.g. a DIP, SO, DFN, etc), and your netlist is created/updated accordingly for the new footprint - the point being, the mapping from one to the other is deliberately done using "pin numbers" and "pad designators".  Altium then by default shows pad-designators and NET LABELS, the name given to the wire between 2 pins on the schematic.

I think I know what you're asking, though, and despite the fundamental design of Altium (and AFAIK most other EDA packages) not facilitating this transference of schematic-symbol "pin names" across to footprint pads, there is a half-way to get what you want:  Assign a net-label to each net that's the same as the pin-name of the schematic symbol.  Of course, the question then becomes, "use the pin-name of which of the two devices!? (in the case they're different)", which is why I called it a half-solution, although you'd still face this conundrum even if you could get pin-names to be shown on pads.

This is why deliberately assigning a net-label (e.g. "SCK", "MISO") to the wires in the schematic can be so helpful for PCB routing - you can see exactly what the connection is for, treat it appropriately, even though the pin-names on the schematic symbols aren't visible in the PCB.