How to correctly use a set of pogo pins?
There are three parts to your puzzle

Electrical Connection

While you can solder directly to pogo pins, they are wear items and easily bent or broken, so serious production fixtures tend to use pogo pin sockets in which the pins are replaceable.  These can often most easily be sourced by doing a search on the part number, for example a P75 pogo uses an R75 socket.  There are at least two types of sockets - solder tail, and what is basically a wire wrap post (or if spaced in the right array, possibly a header pin).  Pogo sockets have a ridge on them and so need to be inserted into whatever holds them from the pin side - which means that if you pre-wire them you need to thread the wire through the hole and then insert the socket, so you have to do that before putting a connector on the far end of your wire or combining it into a bundle.  You can typically make the solder joint small enough, thread the holder through the fixture, and then put a piece of heat shrink over the joint; you can also buy sockets prewired to lengths of wire.

Mechanical Mounting

Production fixture traditionally use holes CNC drilled in a piece of acrylic or similar plastic.   If you don't have ready access to that capability, two viable alternatives are using a common thermoplastic filament 3d printer (in which case you will almost certainly have to ream the holes to size afterwards) or using a stack of multiple purpose-made PC boards.  If you go with multiple boards, consider putting a spacer in between so you increase the vertical distance over which you are fixturing the pins (essentially, treat a 3-piece prototype PCB fab as a CNC machining service that EE's are already familiar with sending jobs to, with the added benefit that you can have traces from the pogo plated through holes to a connector which you install on one board in the stack to bring off your wiring harness)

Actuation

Far east factories operating in a semi-manual mode seem to fixtures where the pogos in their carrier are lowered from above via linear bearings riding on rails and actuated by a toggle clamp of the sort you might see on shop cart casters.  In contrast, small batch US makers tend to have the pogos sticking up from a fixture with some alignment pins over which the board is held, possibly secured by a toggle clamp.

For small numbers of pins as you show you can also make up a simple cable ending in a head with a few pins in it, and hold that in place during a brief flashing operation (but such solution probably won't support enough signals for a full test, only programming and say serial output).  Tag Connect offers a pre-made form of this for a few pin counts and spacings, the idea being that you design the board to their pogo head rather than designing a custom pogo fixture for your board.