Do all inductors produce 1 weber after one second when 1 volt DC is applied?
My stab at it (revised). The original block quote :

If you take a loop of superconducting wire, and apply 1V to this wire during 1s, then the magnetic flux inside this loop will have changed by 1Wb.

With qualifications that this is independent of size, shape. material ... but with no qualification about the number of turns. This leads to:
Wb = V * s  ... eq1
It says nothing about the current flowing in the turn (or turns) and leaves unanswered whether an N turn coil obeys
Wb = V * s  ... eq1a
or
Wb = V * s * N ... eq1b
or even
Wb = V * s / N ... eq1c
Note the definition of Weber

The weber is the magnetic flux that, linking a circuit of one turn, would produce in it an electromotive force of 1 volt if it were reduced to zero at a uniform rate in 1 second

(yes from Wiki but that links to a primary reference) so it is the flux related to 1 V-s explicitly in a single turn. A crucial difference of phrasing absent from the linked page...
A second turn in the same field would be an independent voltage source.
This brings the definition in line with eq1c because 1 Weber is the flux related to 1V-S per turn.
So my (revised!) understanding of the original quote is

If you take a loop of superconducting wire, and apply 1V per turn to this wire during 1s, then the magnetic flux inside this loop will have changed by 1Wb.

This supports Andy's understanding of Faraday's Law expressed in the question - to keep the rate of change of flux constant, you need to keep the voltage per turn constant. Alternatively, if you halve the voltage per turn you will indeed halve the rate of change of flux.
It also leads to the modification in Eq1 of the linked webpage. Which then leads logically to his final equation
H = Wb * turns / A
or
Wb = H * A / turns
This originally made me suspicious, because one normally sees flux as proportional to ampere-turns, so amperes/turn looked ... unfamiliar. The reason is that the inductance already contains a turns-squared term :
L = Al * n^2  (where Al is called &quot;specific inductance&quot; and is a constant for a particular geometry and material)
H = Al * turns^2
Substituting for inductance brings us back to the familiar ampere-turns
Wb = Al * A * turns
which is a more convenient form for some purposes in inductor design.