What is the precise definition of Ramachandran angles?
Definition

Protein backbone dihedral angles are called φ (phi), involving the backbone atoms COn-1 - Nn - Cαn - COn , ψ (psi) involving the backbone atoms Nn - Cαn - COn - Nn+1) and ω (omega) involving the backbone atoms Cαn - COn - Nn+1 - Cαn+1). Thus, φ controls the COn-1 - COn distance, ψ controls the Nn - Nn+1 distance and ω controls the Cαn - Cαn+1 distance. Where n is the current atom for which you want to calculate the dihedral angles. Dihedral angle varies between -180 to +180. It is positive if rotation between atoms is clockwise and negative if rotation is anticlockwise.

Source: http://mebinfotalk.blogspot.com/2013/06/how-to-calculate-dihedral-angle-using.html
And here is an illustration from the same source:

To calculate, use the cross product to get vectors normal to the (v2, v1) and the (v2, v3) planes, and calculate angles between them (it is easy to get the direction wrong, so I would test with known cases).
In the question, the OP states:

They rely on the fact that two bonds on one side are in the same plane as the rotating bond.

No, the plane is between the rotating bond and the adjacent bond. Two bonds of one atom always form a plane. Here is an illustration: Dihedral angle of gaseous and crystalline HOOH
3D demonstration
A nice interactive tutorial can be found at Proteopedia: https://proteopedia.org/wiki/index.php/Tutorial:Ramachandran_principle_and_phi_psi_angles
Omega
Omega is the torsion angle of the peptide plane. It is most often near 180 (trans peptide) but sometimes near zero (cis peptide).
In a PNAS paper from 2012 (https://doi.org/10.1073/pnas.1107115108), the authors state in the abstract:

Analyses as a function of the φ,ψ-backbone dihedral angles show that the expected value deviates by ± 8° from planar as a systematic function of conformation, but that the large majority of variation in planarity depends on tertiary effects.

In protein structure determinations, the omega angle is often restrained, especially in low resolution structures. Looking at structures deposited in the protein data bank, it might therefore seem that the distribution of the omega torsion around 180 is more tight (less varied) than it is in reality.
First and last dihedral
The first amino acid has no phi angle and the last has no psi angle. Exceptions are when the N-terminus or C-terminus are modified.
Bond angle tau
Different from the torsion angles phi and psi, the angle tau is a bond angle (depends on three atom position, not four). It is the angle at the C-alpha carbon between the nitrogen and the carbonyl carbon. As an $sp^3$ carbon, the C-alpha carbon is expected to have an angle around $109^\circ$. The larger the angle, the more (phi, psi) combinations are possible without steric clashes, i.e. the larger the &quot;allowed&quot; regions in the Ramachandran plot.