Why do different authors assign different characters to the same orbital or normal vibrational mode? What is the source of these differences?

If different orientations of the molecule are used, symmetry labels corresponding to the same orbital or mode may be different.

For example, for the water molecule which symmetry axis coincides with the Z axis, we still have two choices for the molecular plane: XZ and YZ presented in the picture below.

Two different orientations of the water molecule

The following table shows symmetry labels of water normal vibrational modes for these two different orientations.

Vibrational mode	Irreducible representation for
Vibrational mode	XZ plane	YZ plane
Asymmetric stretch	b₂	b₁
Symmetric stretch	a₁	a₁
Bending	a₁	a₁

The difference in labeling for the asymmetric stretch follows directly from the definition of irreducible representations for the C_2v point group symmetry given in the table below.

Character table for the C_2v point group symmetry

	E	C₂ (Z)	σ_v (XZ)	σ_v (YZ)
A₁	1	1	1	1	z, x², y², z²
A₂	1	1	-1	-1	R_z, xy
B₁	1	-1	1	-1	x, R_y, xz
B₂	1	-1	-1	1	y, R_x, yz

As we can see, B₁ and B₂ irreducible representations differ in the way they change the sign upon reflections in the XZ and YZ planes. Thus, different selections of the plane of the molecule (XZ or YZ) lead to different labeling of the asymmetric stretch (b₂ or b₁).

In order to solve these ambiguity, the symmetry conventions were developed.

What are the conventions for symmetry notations in computational chemistry and spectroscopy?

Conventions of the symmetry notations are given in R. S. Mulliken, J. Chem. Phys., 23, 1997 – 2011 (1955).

The most important symmetry conventions are:

lower case letters should be used to describe normal vibrational modes and orbitals, e.g. a_g normal mode;
upper case letters should be used to describe vibrational, vibronic, and electronic states, e.g. A_g electronic state;
character symbols and their definitions should be used exactly as given in G. Herzberg, "Infrared and Raman spectra of polyatomic molecules" (New York, 1945), besides T and t symbols should be used instead of F and f for triply degenerate species;
planar C_2v molecules should be oriented such that the X axis is perpendicular to the plane of the molecule, and the Z axis is the axis of symmetry;
planar D_2h molecules should be oriented such that the X axis is perpendicular to the plane of the molecule, and the Z axis passes through the greatest number of atoms. If the last condition does not eliminate the ambiguity, the Z axis should pass through the greatest number of bonds. For other point group symmetries, read the original paper;
Normal vibrational modes should be labeled in accordance with the following rules:
1. Modes should be grouped in blocks with the same character.
2. Blocks with different characters should be ordered as in Herzberg's book (p. 272).
3. Modes inside of each block should be ordered from the highest to the lowest vibrational frequency.

What information should be presented to make symmetry assignment unambiguous?

The only way to make the symmetry assignment unambiguous for the reader is either to follow standard conventions (preferred) or to specify the orientation of the molecule in space.