This spigot leaks only when running, from the handle spindle. I have never worked on one like this, but I assume the handle needs to be removed with the front screw, then the packing nut needs to be removed. After that, new packing can be placed on the inside of the packing nut and retightened. Is this correct? Is that the type of packing that will be required?
First, hold the body of the valve still with one wrench, and tighten the packing nut a bit with another wrench. If you can snug it up and stop the leak, while still being able to turn the handle, that's all you need to do.
If that fails, you seem to have the correct idea for repacking. Do remember to turn the water off ahead of this valve or it can get very exciting when you remove the packing nut. Generally the packing (modern) may be white teflon or may be black from graphite additions as you show. It doesn't matter much. Old packing was various other things, depending on how old.
Might want to start with putting penetrating oil on the handle screw, it looks like the typical rusted-in-place likely-to-break screw we see there too often. When you get it out, consider applying anti-seize compound to a replacement, and consider a non-corroding screw for a replacement.
