What makes you think that defining a magnet as "something with a magnetic field" is a recursive definition? that would be true if the definition of magnetic fields was "something a magnet has", but that's not the case. Quite simply, you CAN define magnetic field without defining a magnet, but not vice versa. Read the definition of magnetic field:
https://en.wikipedia.org/wiki/Magnetic_field
In physics, a magnetic field is that part of the electromagnetic field that exists when there is a changing electric field. A changing electric field can be caused by the movement of an electrically charged object, as in an electric current; or a combination of the orbit of an electron around an atom and the spin of electrons themselves, as in a permanent magnet.
if you are really concerned about having a single definition, it really shouldn't be too hard for you to combine those two definitions together.
It seems your concept of what a definition may and may not contain is skewed. By your logic, it would be improper to define a light bulb as a device that uses electricity to produce light, because it has the word 'light' in it, or that you have to know what a light bulb is to define light itself... and that makes no sense.
Guess what, it's pretty common to define things in terms of other, established definitions, otherwise every time you needed to define something, you'd have to explain half of physics to go with it. If someone does not feel that the definition is sufficient, then they have the freedom to look at the definitions of the concepts in terms of which the original thing was defined. For example, if you don't feel that "something with a magnetic field" is a sufficient definition of a magnet, you are free to look at the definition of magnetic fields. and if you don't find that definition sufficient, you can then look at the other things used to explain it; electromagnetic fields, electric charges, current, electrons, atoms, etc. and then you can keep digging down through an ever-expanding heirarchy of definitions and explanations until your head explodes, or you reach the end of the current knowledge of physics, whichever comes first.