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I don’t see a conflict between those two statements. Functions are special cases of partial functions: they are the ones that are entire as relations.
“a partial function is the same as a functional relation seen from a different point of view.”
The understanding problem may involve the phrase “is the same”.
A partial function is a generalization of an “entire functional relation” in that it is not necessarily entire. However a partial function $f: A \to B$ is also equivalent to a total function to $B + \{\top\}$ which is a pointed set where the added point $\top$ is the “trash” point which stands for “undefined” or “unknown”. This added structure is a specialization.
Thus the relation between partial functions and total functions can be described as generalization, equivalence, and specialization. I expect the equivalence expresses a duality.
I think one could multi-point $B$ above with distinct points, $\top_1$, $\top_2 ...$ for each unknown value where the different $\top$s are isomorphic. Something like this is needed if one wants to capture the ordering of partial functions - if $f$ is a functional relation and $g = f + \{a \mapsto b\}$ augments $f$ with a new mapping that was undefined then $f \lt g$.
Normally this extra point is called $\bot$ rather than $\top$ because it's at the bottom rather than the top of a partial order. This makes $f \leq g$ automatically work out in your example as you want: $f(a) = \bot \lt b = g(a)$. We should say more about this ordering at partial function.
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