When you compare, you say that A is better than B, which inherently implies that B is worse than A.
The possibly taken offense you're trying to prevent is the inference that because A is better than B, that makes B objectively bad (or not good enough).
That is not strictly the case. An elephant weighs more than a cow but that doesn't inherently mean that a cow is a lightweight. Similarly, you might not like your old name as much as you like your new name, but that doesn't mean you hate your old name.
I'm aware that you in fact do not like your old name, but that's not a given. You might simply like your new name more.
So the simple solution is to not compare them. Instead, simply state that you've started using the name and have grown to like it.
Q: Why did you put this name on the mug and not yours?
A: I've been using this name for a while now and it's grown on me.
Don't open the door to asking people if they like the new name, because that gives them a platform to disapprove of it. Instead, simply state it as a fact that it is a name you use (in some circles), and that you chose to do so.
Don't make any comparison with your old name, and do not confirm/deny any comparison made by others.
If they dig down into why you decided to change your name, you have the benefit of claiming you initially did it for privacy reasons, since you started using it online. It's not the truth, but it hides the awkwardness that comes with openly disliking your given name.
If you do want to change your name to your new name, and want your family to use that new name, I suggest broaching that topic in the future, when they are already aware of the second name and have had time to get used to it.
Based on the comments, I feel like you're not sure what you want to do. You're trying to broach the topic of your name change, you don't want to tell the truth, and you don't want to lie either.
The only remaining option left is ensuring that your family does not respond altogether, but shutting people up for what amounts to an innocent genuine question is not a good interpersonal approach.
- If you want to avoid lies (including white lies or misdirections) at all costs, and don't want to tell the truth either, then the best approach is to keep it hidden.
- If you want to broach the topic but want to avoid lies (including white lies or misdirections) at all costs, then you're going to have to be tell the truth.
- If you want to broach the topic, but prefer not to tell the truth when asked about it, then you're going to have to resort to white lies or misdirection.
When considering what you're trying to achieve, I think the last option is the most desirable. White lies are not morally reprehensible (it's an essential part of civil conversation) and are the best way to broach the topic while avoiding an awkward conversation.
You actually have really good explanations for this new name that answer the question of "why?" without needing you to mention your dislike of your old name:
- Since you use this new name online, you can argue that online usernames are often aliases, for privacy's sake.
- Since you use this name with your friends, you can argue that it's just a nickname that grew over time.
I have some personal experience with the nicknames, and it's perfectly acceptable for a group of friends to evolve into having nicknames for each other. Not all nicknames will stick, but some eventually will, and you'll end up using the nickname unironically.
Even if your family doesn't like your new name, or doesn't understand why friends would use nicknames; neither is a valid argument as to why you can't use this nickname with your friends. You mentioned that your family has a nickname for you as well, and they wouldn't accept an outsider telling them they're not allowed to call you by that nickname. Similarly, they can't tell you or your friends that you're not allowed to use the nickname you want to use.
In either case, this sets you up to slowly start using this new name more and more, and eventually have it be the name people call you by. And you wouldn't even need to lie about it when questioned, because what you say is the actual literal truth:
I have come to really like this name and want to keep using it.
The core point remains the same: don't compare the new name to the old one, because that invites a discussion on why you don't like your old name (as much). Simply talk about liking the new name and your intent to want to keep using it.