In short:
* The angular momentum (about the origin of the coordinate system) of a particle that moves along a trajectory $r(t)$ is $L=r\times\dot r\,.$ For simplicity lets assume the particle has mass one. When we have several particles (not necessarily part of a rigid body) their total angular momentum is $L=\sum_ir_i\times\dot r_i\,.$ This is your first formula. It is very general as it does not require a rigid body and holds for the points performing any type of motion.
* The angular velocity $\omega_i$ of particle $i$ is not so easy to define in general. It requires to distinguish between translational and rotational motion. In other words: it requires to extract from $\dot r_i$ the component that is not translational. My favourite book on that subject [1] achieves this by defining the socalled transferred velocity (p. 499
125). To do this a moving coordinate system is introduced. In the case of no translational motion the angular velocity $\omega_i$ is defined by $\dot r_i=\omega_i\times r_i\,.$ A priori it depends on $i\,.$
For simplicity I assume not translational motion from now on.
* When the particles all are part of a rigid body one can find an angular velocity $\omega$ such that they all satisfy $\dot r_i=\omega\times r_i\,.$ This angular velocity does not depend on $i\,.$
The total angular momentum of the rigid body is then: $$ L=\sum_ir_i\times(\omega\times r_i)=-\sum_ir_i\times(r_i\times\omega)\,. $$ Another important step is now that this equation should be formulated in a coordinate system that rotates with the body: $$ L^{\text{body}}=\sum_iR_i\times(\Omega\times R_i)=-\sum_iR_i\times(R_i\times\Omega)\,. $$ See [1] for details. Using the trick $$ a\times b=\left(\matrix{0&-a_3&a_2\a_3&0&-a_1\-a_2&a_1&0}\right)\left(\matrix{b_1\b_2\b_3}\right) $$ it is fairly easy to see that the linear map $\Omega\to L^{\text{body}}$ is given by a symmetric matrix $I$ which we call inertia tensor of the rigid body. In other words, the formula $$ L^{\text{body}}=I\Omega $$ holds under all assumptions made so far.
Does this help to rethink the subject? I have the impression that to answer all your questions it is inevitable to go into many details that [1] has laid out much better.
[1] <PERSON>, Mathematical Methods of Classical Mechanics. 101