Outer wrap has everything to do with pulling tension on the instrument. If it didn't, there would be no point in making wound strings in the first place.
In reality, since plain steel strings become too stiff and inharmonic past a certain gauge (available plain guitar string gauges stop at 026p), we do make wound strings, which are more flexible than solid ones, but which can be tuned to lower pitches (than their bare cores would be) and not flop out because they're more massive (than the aforesaid bare cores), so their usable tension range is decently high for a stable pitch to be attained with normal plucking force.
And why so? why is their tension higher than it would be if only the core were there, no wrap?
Because both core and winding contribute to tension at pitch, since it is the total mass that counts, not just that of the core.
(By the way, the thickness of wound string cores is exactly that of plain steel strings: high F bass strings have cores that are as thin as light-gauge high E guitar strings; cores of very large strings like 160 or 175 have 024" or 026" thick cores. The difference is just that most cores - but not all - are hexagonal rather than round-section.)
The core is the one element in a wound string's construction that bears most of the tension (the windings are only under a fraction of that tension); however, this is different than saying it is the sole element that determines it, which it is not: it is instead, as I said, the total mass of the string, including core and winding, that matters (scale and pitch being equal, of course).
The DR salesperson email in post #13, beside not addressing (whether intentionally or not) the tension question asked by that poster as well as this thread's OP, is not in support of the claim that only the core matters, either: it just mentions that, prior to winding, the pretensioning (without which winding could not happen) of the bare core is made at the same tension value for all string models.
(If you've followed what I've been saying, you'll correctly guess that, once the winding is complete, the tension, without any adjustment imparted to the string by the machine, will have nonetheless significantly increased, and this precisely because of the added mass of the winding alone. That's why they say "before the wind".)
However, nanbanpapa's reasoning on winding material making a measurable difference only works, in terms of providing an easy-to-read spec useful in predicting tension differences, provided everything else is equal, which we cannot be sure about.
Are diameters of core (as FRoss6788 points out), inner and outer winding wires the exact same? Only in that case will a pure nickel winding be the sole element determining a higher tension than that of an otherwise identical, steel-wound string. And of course, nickel-plated will make even less of a difference.
(A propos of which, the steel under the nickel is itself probably different than the steel used in stainless steel strings.)
Now, put an even heavier metal in the mix: lo and behold, they use copper in classical guitar and piano strings. What about an even heavier metal, so as to get things really cooking? Yep: they do use tungsten in double bass strings, and not because some mysterious tonal quality, but because it is heavy as heck.
