I'm not a beginner. However, at one time I was and have had many teachers along the way. I'll try to answer by recalling my thoughts at that stage of learning music.
I started piano as a child, and one of my first experiences was a group lesson with other children. The teacher was informative, I learned some basic stuff, but what I was looking for at that stage was engagement. Doing the method book thing was interesting, but I wanted to know how make music, how to take all those symbols on the page and patterns learned by rote and turn them into music. Somehow I sensed that the lesson plans from this teacher were not for me, just not the right fit. I sought something more.
Still as a beginner, I found a great teacher that instilled a great love of music making. Everything she did was musical: the way she talked, the way she used rhythm in her speaking, the manner she sang to demonstrate a phrase, the energy she used to teach a theoretical concept, she would move her arms and dance in order to demonstrate a topic. Even something as mundane as music symbol flash cards were musical, not because the stuff printed on the cards were music symbols, but because the presentation exuded a love of music. At this stage of my learning, I think this teacher found the right balance of motivation and teaching beginner fundamentals. We were still using a lot of method books, but a good amount of lesson content and creativity came from the teacher herself, the books just provided the sheet music. Unfortunately, this teacher moved away.
Sometime around an early intermediate learning stage, I had an increasing interest in classical music and wanted to focus on that. What I really needed at this point was a teacher that knew a lot about technique. One teacher I had during this time knew some stuff about technique. It was a very cookie cutter approach, with every student in the studio basically following the same plan and playing the same pieces regardless of musical interests. The kind of teacher that makes every student follow the same fingering, regardless of differences in hand shapes. It was technique taught by the book, literally. From my perspective, interesting at first, but looking back, potentially dangerous in the long run as this teacher did not do enough to correct flaws and inefficiencies in technique that could lead to injury with more advanced playing. While this teacher was able to get me out of method books and into some real repertoire, there were gaps in teaching a sufficient technique to support that repertoire, and more importantly to support healthy playing habits. As a student, I felt limited because I was unable to explore repertoire that interested me (the teacher was very resistant to students bringing in music they were interested in), and because the technique as it was being taught felt constrictive - like it did not allow me to express music with greater ease. I thought it was time to move on.
At this point I was starting to get interested in things like the Chopin Etudes. I was fortunate to find a wonderful teacher, a concert pianist, that had a deep knowledge of technique, a technique that enabled expressive capabilities and removal of unnecessary tension, a technique that made playing challenging music engaging. This teacher showed me how to experience technique that was enabling, a gateway to explore music I was curious about... I never knew that the playing mechanism could feel so fluid and free when tackling challenging music. I stayed with this teacher until the start of university studies.
These are just a few of the teacher's I've had, the ones I thought were most relevant to the thread topic.