In Lecture Notes in Algebraic Topology by Davis and Kirk it seems that given a (convergent, bigraded) spectral sequence $(E^r, d^r)$ one defines the $E^{\infty}$-page as $$E^r_{p, q} = \operatorname{colim}_{r \to \infty}E^r_{p, q}.$$
However it seems that the convention (at least by books such as Weibel's: An Introduction to Homological Algebra and Rotman's An Introduction to Homological Algebra) is that for any (bigraded) spectral sequence $(E^r, d^r)$, one defines $E^{\infty}_{p, q}$ in the following way. Suppose that $d^r$ has bidegree $(a, b)$, then we define $B^r_{p, q} = \operatorname{im} d^r_{p-a, q-b} : E^r_{p-a, q-b} \to E^r_{p, q}$ and $Z^r_{p, q} = \operatorname{ker} d^r_{p, q} : E^r_{p, q} \to E^r_{p+a, q+b}$ so that $$E^{r+1}_{p, q} = Z^r_{p, q} / B^r_{p, q}.$$
Then one defines $$E^{\infty}_{p, q} = \frac{\bigcap_r Z^r_{p, q}}{\bigcup_r B^r_{p, q}}.$$
Now at some point in Lecture Notes in Algebraic Topology by Davis and Kirk it seems that implicitly it is the case that $$E^{\infty}_{p, q} = \bigcap_{r}E^r_{p, q}$$ as in one of the proofs it is stated that $E^{\infty}_{n, 0} = \bigcap_r E^r_{n, 0}$, which leads me to guess that the above holds.
More confusingly still, it seems that in an earlier version of Lecture Notes in Algebraic Topology by Davis and Kirk, the $E^{\infty}$-page was defined as $$E^{\infty}_{p, q} = \lim_{r \to \infty} E^r_{p, q}$$ as indicated by this question here on Math.StackExchange: definition of convergence of a spectral sequence
So it seems that there are possibly four ways of defining the $E^{\infty}$ page of spectral sequence, in what ways are these equivalent? Do they agree for convergent spectral sequences?
Also I don't see how the authors could swap out taking a colimit for a direct limit as categorically speaking those should be dual notions.
Also I'd like to mention that I am inferring a lot from the text, I'm sure that it is due to a misinterpretation on my part, (and not the authors) that I've claimed something incorrect to hold.