Bad first month in my PhD

It's your first month. You will be in the program for 3-5 years. Everyone knows this, and you will have the opportunity to redeem yourself. Be patient. Do better in the future. Everything will work out alright.

What's more important is that you understand your weaknesses. Work on them. Become a better person and scientist by figuring out how you can address your weaknesses by either addressing them directly, or working around them.

Now I am facing the problem that it will be harder for me to build trust with my advisor. What steps could I take to start building it again?

Fix the problem. If your advisor's getting a bad impression, that's a symptom; like a cough, you can try to suppress it, but the long-term solution is to fix the cause, i.e.:

It seems that I really didn't understand concepts.

So, sure, some damage control can help for now. But the long-term priority needs to be personal improvement.

All that said, it sounds like you're beating yourself up a bit. That's not necessarily a bad thing; being critical of oneself is a great way to avoid becoming complacent. But at the same time, make sure that you don't overdo it and get depressed. There'll need to be a healthy balance.

Like @WolfgangBangerth said, you'll be in the program for some years yet. You probably have a bit of time to breath and figure things out.

There are occasional shakeups early on in a PhD. You may voluntarily or non-voluntarily switch groups. That may add time to your PhD. But it's going to be OK. I have friends with similar experiences who got kicked out of some hotshot's group, and they had very positive experiences with a different advisor. They have cool jobs now to boot.

The first year of a PhD can be brutal. You can go from the top of the undergrad class to the middle of the pack, or even the lower half. But don't fall into the cognitive trap of downward comparison: searching for someone above you and comparing yourself to him/her. This is virtually ALWAYS possible no matter how smart, rich, or strong you are. After all, even those achieving some maximum hold it only temporarily. We are simply points in a high-dimensional distribution of abilities and it's nonsensical, non-healthy, and immodest to consider yourself the absolute max or deserving of it.

It sounds like your anxiety and self-shaming up until now have helped motivate you to do better. And to some extent, that's a normal psyche in a PhD student and many over-achievers. But now you would do well to reign in those strategies and add balance to your life. There is more to learn than best coding practices in your PhD; there is also all the emotional management and self-care skills that will afford you the peace of mind you need to code well.

In terms of not taking the initiative, that sounds related to shame and anxiety as well - You may want to do "right" according to someone else's definition of what right is. Let this opportunity challenge that approach. So much of a PhD is about learning to think and do independently.

Go with confidence!