Let me say this right now - your IP was privately checked by a third party.
It wasn't me. Non-malicious, and your IP is still being kept confidential. Even I don't have your IP.
The IP was gained through a website that you signed up on - so yeah, it's not like your IP was hidden. Plus, you signed on very recently before the IP-check (after you posted this thread), so...
What I'm told is that the distance between the two IPs was very minimal - almost nonexistent, aside from the fact that "Nirmit" came on using a wireless connection. In fact, you're currently using the same IP.
Also, how did you become aware of MCGamer and Badlion? I can understand if you knew of MCSG before your account was stolen, but Badlion rose to prominence relatively recently.
Also, if you want to be unbanned, the staff team is going to have to do an IP-check. If not, you and your other accounts are staying banned. Only people involved in the checking process (Senior Moderators or higher, all of whom are very trust-worthy) will have access to your IP, and only for the time that they need it. If you're okay with leaving the MCGamer community, then you don't have to under-go an IP check.