I have in my understanding that "Mode" is just a path of light propagation in a fiber.
That's a very rough way of understanding it. A better way to describe it is that a mode is a way for the EM field to satisfy the boundary conditions imposed by the physical structure of the fiber, such that the field patterns at different positions along the fiber's length remain the same.
If light propagates in an Optical fiber in a single path only, then its a single Moded fiber and the Mode count is 1. If there are multiple paths in which light can propagate, then there are multiple modes.
Again, roughly correct.
If there's only one field pattern that satisfies the boundary conditions and propagates unchanged along the fiber, that's a single-mode fiber. If there are multiple such patterns, then that's a multi-mode fiber.
and if its correct, how does this mode relate with TE and TM modes?
First, TEM modes only occur in waveguides with multiple conductors (like coaxial cable). Optical fiber is a dielectric waveguide, so there is no TEM mode.
There are TE and TM modes.
In multi-mode fiber with high mode counts (10's or 100's of propagating modes are typical in the most common multi-mode fiber types), there are also various other modes with more complex patterns. Generally we don't have to know the exact field patterns of the different modes. Instead we use empirically known properties of the modes to predict the fiber's behavior (for example, multi-mode dispersion or coupling between modes due to a change in the fiber geometry).