Yes and no.

When you're first learning a spell, it tends to be important, along with pronunciation and enunciation.

The more points of focus (wand, intent, the way you're saying the spell name) you have, the better your chance at decent success instead of something mediocre or that doesn't work at all.

The thing about a wand, however, is that it's a focusing tool. A specalised focusing tool--which is why they're often so highly specific to their owner or why certain combinations work better for certain types of magic--but still only a focusing tool.

Hedgewitch wands also work well in a pinch as they're a familiar shape of focusing tool to the point that it ends up not mattering all that much that they haven't got a core.

A wand, however, isn't strictly necessary for a lot of things as evidenced by the fact that most people can do at least some level of wandless casting and that children are able to produce magic without a wand (not always well or necessarily controlled, but they can do it).

After awhile, it becomes second nature to use it that way and you're not necessarily even aware that you're focusing through a wand; it's also sometimes why you'll see someone who is perfectly capable of casting, say Lumos, wandlessly still make the motion they'd make with a wand with their finger or their entire arm.

Habit.

There are some wand woods that lend themselves to silent casting a bit easier (Alder being one of them. Alder is amazing for that sort of practise; if you're having difficulty with silent casting, get an Alder wand to ease into it then step back to your regular wand), and there are some spells that, once you've learned them, don't require much more than passive focus.

Most utility charms are like that and most adults can cast them silently and wandlessly.

The main reason you see people start to get a bit shouty when they want to make sure what they’re casting is as powerful as possible is that doing that adds an additional level of focus to the cast.

The more difficult a spell is, the more focus it takes, and the more difficult it becomes to cast at all if you:

- Don't use a wand.- Use a wand but are sloppy with the cast pattern.- Try to cast it silently.- Mispronounce it, even a little.- Have wavering intent.

Trying to use a spell in a way that goes right to the edge of what its original intent was also makes it significantly more difficult to cast.

Going back to Lumos, you can use it to light a fire if you're able to force it into compliance because it--counts as within the boundaries of the design, it's "lighting" something, just not in the way in which it was originally intended. It will resist and if you’ve not done it before, likely not work at all, but you can force it.

That all aside, once you've got something down wordlessly and wandlessly, you can make up whatever words and motions you want to go along with it and it's not going to matter because it's a superficial display and you're actually doing the real casting silently and without your wand.