I watched this video about inertial vs non-inertial frames, at 3:30, professor say particle has zero net force ($\mathbf F=\mathbf 0$), but non-zero acceleration ($\mathbf a≠\mathbf 0$).
But particle just appear to have $\mathbf F=\mathbf 0$ and $\mathbf a≠\mathbf 0$, truth is that particle has $\mathbf F=\mathbf 0$ and $\mathbf a=\mathbf 0$?
Link information:
channel: HC Verma, www.youtube.com/@hcverma2928, title of video: Inertial or Non inertial
In a non inertial reference frame, Newton's laws do not hold. Usually we choose to invalidate Newton's third law and use forces like the centrifugal force, Coriolis force, etc. so that Newton's second law still works. In that case, if the net force is $0$ then the acceleration in the non inertial frame is also $0$.
However, if we want the net force to only include forces that adhere to Newton's third law, then we invalidate Newton's second law instead, and in that case we can have cases where we can have acceleration without a net force.
For example, let's say you're in a car and right after you throw a ball in the air the car slams on its brakes, accelerating relative to the Earth that we tend to take as an inertial reference frame. In the frame accelerating with the car, the ball accelerates forward toward the front of the car. You can either say there is a pseudo-force that exists due to the car's acceleration, so then the acceleration of the ball is explained by that force. Or you can say there is no net force, but an acceleration of the ball was still possible due to the acceleration of the non inertial reference frame.
In classical physics we usually define position w.r.t. a point (origin) and - possibly, if orientation matters - w.r.t. a set of independent directions. Velocity and acceleration of a point is defined w.r.t. a point as well.
Now, once recalled that, it's possible to define a class of inertial reference frames as those frames where Newton's 2nd principle holds in the form
$$\dfrac{d}{d t} \mathbf{Q} = \mathbf{F}^{ext} \ ,$$
being the momentum of a point mass $\mathbf{Q} = m \mathbf{v} $, being $m$ the mass of the point and $\mathbf{v}$ the velocity w.r.t to the origin of the (inertial) reference frame. The class of the reference frames can be defined starting from one of them: other reference frames have the origin in uniform relative motion w.r.t. to the origin of the known inertial r.f., and with no relative time-dependent rotation (or with constant orientation). I'm not treating rotation here for simplicity. If you ask yourself "how can I find the 'first' inertial reference frame?" there's a section later.
Just as an example, and to be more explicit, given two inertial r.f.s with origins $O_a$, $O_b$, the velocity of the point mass $P$ w.r.t. the origins are
$$\begin{aligned} & \mathbf{v}_{P/O_a} \\ & \mathbf{v}_{P/O_b} = \mathbf{v}_{P/O_a} + \mathbf{v}_{O_b/O_a} \ . \end{aligned}$$
For a system with constant mass (let's avoid introducing other complications here, and avoid treating open systems) $m$, and for 2 inertial reference frames with the origins in relative uniform motion, $\mathbf{v}_{O_b/O_a} = \text{const.}$ , and thus $\dfrac{d}{dt} \mathbf{v}_{O_b/O_a} = \mathbf{0}$, Newton's 2nd principle of classical mechanics has the same expression
$$\mathbf{F}^e = m \dfrac{d \mathbf{v}_{P/O_b}}{dt} = m \dfrac{d \mathbf{v}_{P/O_a}}{dt} + m \underbrace{\dfrac{d \mathbf{v}_{O_b/O_a}}{dt}}_{= \mathbf{0}} \ .$$
What's a true force? Broadly speaking is "something" whose action on a system may change its momentum, and can be traced back as an effect of one of the "fundamental interactions": in classical mechanics, these interactions can be qualitatively summarized as gravitational interaction (governed by universal gravitation by Newton), electromagnetic interaction (governed by Maxwell's equations, possibly in the low-speed limit) and contact interaction as a result of microscopic interactions of molecules of "touching" bodies.
With a bunch of accelerometers (or with just one measuring both linear and angular acceleration) or dynamometers/balances, you can measure acceleration (and assumed forces) or forces-moments with these instruments. These instruments are sensible both to "true forces" and "fictitious forces" that may appear in a non-inertial reference frame.
If you can control and accurately know the physical processes producing "true forces", you can compare the expected forces (and moments) with the measurements of your instruments: if they match, it may mean that the "fictitious forces" are zero and the instruments are stationary in a inertial reference frame.
How to deal with non-inertial reference frames, then? If one knows the motion of the reference frame used for the description of the motion, w.r.t. an inertial reference frame, one possibility is to use relative kinematics to transform Newton's second principle for the inertial reference frame for the non-inertial reference frame to get the dynamical equation (usually some extra terms, like Coriolis, centripetal, and other terms appear: I've already answered similar questions before, see here), and the laws of composition of velocities and position.
