I would not recommend a personal re-cell operation. There's many things that can go wrong if not done perfectly right. As you note, solder tabs that are already pre-welded onto the battery terminals are good for minimizing the inherent danger in overheating Li-ion cells if one attempts to solder wires directly to the ends of the battery. Not to mention that most OEM Thinkpad battery packs have built-in protection circuitry that will essentially self-destruct if it notices that the cells have been disconnected from it. Think of a fuse that works in reverse - blowing out if no
power is applied.
As for commercial re-celling, I've had experience with only one: batteryrefill.com If you use this site's search function, you should find some of my posts. They are fairly expensive, running more than the cost of an original battery but the offsetting reason is that they use newer, higher capacity cells (i.e., longer run time). The resulting visual aspects of breaking apart a pack and then gluing it back together can be minus for some people.
Buying aftermarket laptop batteries can be a hit-or-miss proposition and like all Li-ion batteries, one is taking a chance that it won't explode sometime down the line. $30 battery packs must, by definition, cut some corners in order to be profitable at that price level. I understand that potential and have accepted the risk. A recent news story that I read was about a woman who was wearing her personal battery-powered headphones on an airplane. They caught fire and she suffered burns to the side of her face and hands. No details about the manufacturer of the headphones or batteries, but one must assume that they were lithium ion. Recent history is a guideline: Samsung Note 7 smartphones catching fire and being withdrawn from the market (full disclosure: I'm an employee of Samsung but have nothing to do with the smartphone or battery divisions); multiple makers of 'hoverboards', those silly Segway-like rolling emergency room patient generators that blow up when people are riding them, being charged, or simply sitting in a closet doing absolutely nothing.
ref: https://www.wired.com/2017/03/dont-blam ... explosion/