There is strong support for house-ruling that "spectral" implies "magical", based on how the word "spectral" is used in 5e
The accepted answer to this question about Spiritual Weapon states that there is "a decent argument for 'spectral' being equivalent to 'magical force'. However, that determination is up to each table."
Along those lines, the description of Reaping Scythe does not contain the words "magic" or "magical", but generally, effects described as "spectral" in 5e materials are magical. Among spells, here are some examples:
Armor of Agathys:
A protective magical force surrounds you, manifesting as a spectral frost...
Guardian of Faith:
A Large spectral guardian appears and hovers for the duration...
You create a floating, spectral weapon within range...
A spectral, floating hand appears at a point you choose...
...If you are good or neutral, their spectral form appears angelic or fey (your choice).
And generally the word "spectral" doesn't appear to have non-magical application in the 5e sourcebooks. Which makes sense. For what mundane item could be spectral? Therefore, by examination of the meaning of "spectral" in 5e, I would house-rule that attacks or effects from spectral "objects" inherently are magical (it does not matter if we consider them to be attacks or just effects that do damage).
Further rationale for the rouse-rule would be to note that its widely accepted to infer "magical" from descriptions that don't contain the word, all the time. Note that spell descriptions often don't say the word "magic" to describe their effects -- we just take it that to say that a damage effect comes from a spell implies it is magical. In other words, we infer "magic" from the word "spell". The house-rule would be that, likewise, to say an attack is from a spectral instrument implies the same. We would infer "magic" from the word "spectral" because this is consistent with every other occurrence of the word "spectral" across 5e literature.
Besides spells, some class archetype features introduced in XGtE introduce "spectral" abilities that have effects it's hard to imagine not being magical, like passing between planes with the Horizon Walker's Spectral Defense or summoning spiritual warriors with the barbarian's Ancestral Protectors.
But again, none of these say literally that "what is spectral is magical", so you could call it a house-rule (that I would happily make at my table).
Based on this, the answers to your specific questions would be as follows:
The Werewolf would take the full 11 damage, since the spectral slashing is magical.
The Black Pudding has immunity to slashing damage, period (regardless whether magical or not -- if it was only immune to non-magical slashing damage, it would say so), so it takes only the 4 necrotic damage.