There are several ways to do leads and all have to be weighed to determine which is best. I posted somewhere on this recently but can't find it so I'll give my thoughts again.
The traditional lead wires can be effective depending on the situation, type of tinsel lead, where it exits the cone, and how much excursion the driver has. For lower excursion drivers it works just fine almost any way you do it as long as they are the right length. For higher excursion drivers when we do standard tinsel leads we use a thick braided lead, then twist them in a certain way that they swing out to the side and loop just before going to the terminal. Someone else mentioned the twist also. It works good, but can't be used in all situations.
With an aluminum cone, you simply can't go through the cone. It's conductive and you'd short out the leads before you ever got to the coil. Options then are for woven in, like TC used, and sewn on. The woven in has a few issues up front. First, the lead is in the cloth, prior to the cloth being formed into a spider. This creates a couple problems. The cloth is formed by heat, and heating up the lead wire will make it much more brittle in the future. The leads also get stretched as the cloth is formed into the spider because the diameter of the lead is more than the thickness of the cloth. This means the tinsel is stretched and when pressure of the mold is let go it compresses back deforming the spider shape. It doesn't always deform the same and creates unequal lateral force on coil which can make it more likely to rock sideways. Also the leads get pinched with more pressure at the apex of the corrugations making them even stiffer at that point and giving them a natural "breaking point". With any rigid piece of metal, which the lead becomes, you bend back and forth and eventually it will break, or in this case the tinsel lead frays. As it frays and you apply high current it will arc, heating more, fraying more, arcing more, and can melt, burn, or even start the spider on fire. Then there is also the supply issue as nobody in the US makes spiders this way.
The next option is to sew on the lead wires. We have chosen to do this. The problem again was supply. Nobody in the US did this either. I wanted my spiders all done in the US to control the quality. I talked one vendor into doing this and they bought equipment for it. We then went and decided to get a sewing machine we could do this with in house also as we have spiders from other vendors as well. The leads are sewn, then we can pull them to give the right amount of slack to allow for full excursion and to make sure no additional lateral force is put on the coil. The drawback to this is simply cost in labor to do this. It seems to be the best option though as you don't get lead slap, no extra forces put on the coil, the leads stay fully flexible, and you make the connection without going through the cone.
Someone else commented about woofers that only had 2 spiders that should have had 4. There are only two reasons for multiple spiders in a properly designed woofer. To get different compliance values to raise the suspension stiffness, and to support an EXTREMELY heavy coil so it doesn't rock side to side. In the second case there must be space between the spiders to have much effect. Stacking 4 spiders together IMO is pointless when a single spider of equal stiffness to those 4 is much more practical. It is lower cost, and stacking multiples directly together causes them to wear on each other and shortens the life of the driver.
To support a normal coil from rocking, simply putting space between the spider and coil as they glue to the VC form is the best option. For an easy illustration, take a 4ft 2x4. Hold it at one end with both of your hands touching each other and try to keep the other end from dipping and it's hard to do. No matter how strongly you hold you just can't support the whole piece easily from one end. Now move your hands so they are 2ft apart and you can hold it just fine. This is the same thing with VC's in a woofer. Most people expect to glue the cone and spider to the VC at the same spot, far up on the top of the form. All the weight is on the other end though that isn't being supported. Moving the cone and spider even 1/2" apart greatly helps to distribute the load. If doing multiple spiders, they should be evenly spaced to distribute the load between them.
John