Good question . . .
"Light quality" is very subjective, and pretty meaningless really, we can't measure it, we can't quantify it and I don't know what it means.
But, in general terms, I would say that modifiers make a massive difference to whatever light quality is, and the actual light can make some difference, so it's very worthwhile to buy modifiers that are properly designed, properly made and are made from the right materials. Softboxes are products that vary enormously in the quality of their light, for example, and I wrote a tutorial explaining this here
https://www.talkphotography.co.uk/tutorials/which-is-the-right-softbox-for-you.154/
But there are differences between flash units too, although nearly all current flash units are pretty good compared to those of many years ago.
But there are two very different operating systems, and the old one, which basically relied on a combination of switching out capacitors and using a potentiometer to reduce power output, can cause pretty dramatic variations of colour temperature at different power settings, so if we include colour temperature - and we should - as a quality element then basically we should avoid this type if we want consistent colour temperature, and there can also be wide variation in power output too, between flashes, and especially when the power is turned down low. These flash heads tend to be at the cheaper end.
Many are now IGBT - insulated-gate bipolar transistor - and all modern flashguns and most of the better studio flash heads use this technology. Basically, regardless of the power setting, the flash always fires at the same power, and so is always fired at the same voltage, which produces extremely consistent colour temperature. Turning down the power, say from full power to 1/128th power, means that the 1/128th power flash only lasts for 1/128th as long as when it's at full power.
There are a tiny number of really terrible flash heads still around, the price should give them away but in case it doesn't, just don't buy anything that has a fixed, non-removable reflector.
You mentioned systems; flashguns and studio flash. Both have their place, but studio flash is much better in the studio, mainly because they are specifically designed to be used with modifiers. Most (not all) flashguns have a fixed reflector, which makes them usable but far from ideal for use with most modifiers.
Does this help? If not, come back with more questions
