What this question is about (the 4 nomenclatures in short):
Nielsen, M.A. and Chuang, I.L. state in their famous book (see http://dx.doi.org/10.1017/CBO9780511976667, p. 87 ff.) that there are 4 alternative definitions (or variants, nomenclatures) of projective measurements.
1)) Usually they are defined using an observable (a hermition operator) on the state space with a spectral decomposition $$M= \sum_m mP_m,$$ where $m$ are the eigenvalues and $P_m$ the projectors onto the eigenspace of $M$ with eigenvalues $m$. This can be found on page 87 and 88.
2)) Now, the authors also mention that measurements alternatively (with an alternative nomenclature) are defined by giving a basis $|m\rangle$ - which is nothing else but giving $P_m$. This can be found on page 88-89.
3)) The third variant apparently is to give a complete set of orthogonal projectors $P_m$ that satisfy the completeness relation $\sum_m P_m=I$ and $$P_mP_{m'}=\delta_{m,m'}P_m.\tag{1}$$ This can be found on page 88-89.
4)) Moving on they also say that with $\vec{v}\in\mathbb{R}^3 $ and $$\vec{v}\cdot\vec{\sigma}=v_1\sigma_X+v_2\sigma_Y+v_3\sigma_Z$$ one can define an observable (with eigenvalues $\pm 1$), that corresponds to a measurements "along the $\vec{v}$ axis" (which is a historic artifact regarding spins). This can be found on page 90.
The question itself:
How do these nomenclatures / definitions connect? Are they the equal? What is your take on the following thoughts of mine?
My thoughts:
Thought 1: Firstly, what's strange about 1)) in comparison to 2)), 3)) and 4)) is that suddenly the eigenvalues don't seem to be important. This is the first difference I notice, although there seems to be a more relevant one...
Thought 2: Namely, that in 2)) one has more demands on the $P_m$'s (as stated in 2)) above) - they need to be a) a complete set b) of orthonormal projectors and c) obey (1)- It seems odd that 1)) doesn't demand the completeness relation to be true. Truly, on page 88 Nielsen and Chuang do say that 1)) is a special version of a more general postulate, that indeed demands completeness relation to be true. Why don't they then include this in the definition on page 87? Are all these demands taken care of by saying that in 1)) $M$ is hermitian?
Thought 3: Lastly, I see a difference between 3)), 4)) and 1)), 2)). 3)) and 4)) seem to be using a basis of the state space. To see that 4)) uses a basis one just needs to use the usual bloch-sphere representation to understand, that $\vec{v}$ gives an axis in the blochsphere, which can be associated with 2 antiparallel arrows - 2 basis-vectors of the state space. In contrast, 1)) and 2)) don't explicitly say that there is a basis involved. It might be included inmplicitly from what already was topic of thought 3: Maybe the 4 demands on $P_m$ (mentioned in Thought 2) already determine, that $P_m$ looks like $P_m=|\psi_m\rangle\langle\psi_m|$ for a basis $\{|\psi_m\rangle\}$ of the state space.
Thought 4: In my opinion 2)) and 1)) cannot be equal. If you look at 2 qubits and just measure the first one - using the observable $\sigma_z\otimes I$, you surely won't be using a basis of the state space.