I’m going to comment in two parts, starting with human behavioral and then later addressing physiology.

It feels like I don’t have a real routine

I cannot underscore how important it is to develop a routine – any routine – since that’s the trick towards a long-term approach to fitness. Humans are kinda weird in that regard, in that they have to be correctly conditioned to do something week after week. Humans don’t approximate machines at all in this regard, and there’s even less reinforcement when the activity in question does not have a logical structure.

To that end, it would behoove you to formulate a more rigorous routine, one which assigns certain exercises to specific days, ideally occuring at around the same time each day. If after a few weeks (3-4 weeks) you find that some days are too heavily-loaded while others are a cake walk, that’s a cue to increase the sets, reps, or weights for the easy days, and ease up on the same for harder days. Mentally conditioning yourself to the strains of exercise is part and parcel to physically conditioning yourself for the same.

Correct me if I am wrong but I think it could be a better idea to just listen to your body and not to a routine?

It is correct to listen to your body, but the adjustment shouldn’t be to just randomly throw in a rest day. But rather, fine-tune your routine so that your exercise regime matches what your body is presently capable of. As your body improves, adjust accordingly.

My recommendation for picking which days will be what is to form it around your existing work or life schedules. If, for example, leg day always leaves you totally exhausted, then maybe put that for Friday so you’re recovering (physically and mentally) on a slow work day or over the weekend. What order you choose will depend on how you can optimize your fitness plan into your life plan, but whatever you end up with, stick with it for a few weeks and take notes – ie hard copy – about what is or isn’t working.

Which brings us to the physiological side of things.

I came to the conclusion that I absolutely might not need a rest day if I have the energy and motivation to go to the gym.

The present understanding of human fitness is that rest days serve multiple valuable purposes, whether the objective is building muscle, functional strength, endurance, or weight loss. There’s also the matter that – disregarding performance enhancing interventions and their major side effects – the human body cannot totally transform itself in short spans of time, there isn’t much benefit in trying to rush a fitness program. If anything, it tends increases the risk of injury or developing poor form and habits.

With the exception of professional bodybuilders and athletes, whose occupation wholly surrounds their fitness, everyone else can and should take their time to achieve their fitness objectives. To be clear, I’m not suggesting one-month rests between exercises, but rather, the schedule or program adopted should not be unduly swayed by time pressures, instead guided by what your body is saying and what it can do. Skip days can and will happen too, and a few won’t seriously disadvantage you in the long-term. Though a few weeks off could break the fitness habit, and you’ll have to reestablish it.

For the routine you’ve written out, I’m somewhat concerned as you’ve got back exercises on Tuesdays, Wednesdays, and Fridays. That doesn’t leave a lot of days in between to recover. Biceps twice a week seems fine though, since the thinking is that 24-48 hour rest for biceps is acceptable. But seeing as biceps and triceps are part of the same logical body part, while remaining separate muscle groups, there wouldn’t be an issue with having a general “arm day” that focuses on those groups, although some overlap into the other days would be acceptable.

In concrete terms, I would suggest that you orient your fitness routines into a schedule, whether that’s still 6 days a week, or 5 days, or whatever. As general advice, make sure your water and diet intake is alright, and you’re sleeping a decent amount per day.