Sorry, but there is no reason to dermal punch an earlobe when they make needles at that gauge. 8G-6G needles are very common, why couldn't they have just pierced you at that size? Also, a good piercer would know that a taper is NOT good jewelry for healing; they shouldn't even be left in fresh stretches, let alone a fresh piercing.
Not trying to criticize you or anything, but if you had that done without problems that's just luck. It doesn't mean dermal punches and healing with tapers are good for lobes. It's much better to get pierced with a needle and proper jewelry.
The reason dermal punches are so awful for lobes is because they remove tissue, which makes it very easy for your lobes to thin in the future. Piercing needles do not remove any tissue (some people will insist they do but they don't) so you can keep it all for stretching. Punches should only be used on hard tissue like cartilage that can't be stretched easily.
You probably won't run into any problems since your punch was so small and you don't plan on stretching past 2G, but if you decide to go bigger you might notice some thinning. Hopefully everything will go smoothly for you. Pleeeease do not get your other ear punched at 2G... get it pierced at a large gauge like 8G-6G (by a different piercer) and stretch up. If you get that ear punched at 2G when the other ear has only been punched at 6G you'll end up with one ear thinner than the other.
I have 1/2" lobes and 11 piercings: left helix x2, both conches, both nostrils, septum, navel, and VCH.
My lobes are small but I know a lot about stretching and how to do it safely, so message me if you have any questions!