Change in Flux Linkage induce Emf

Mar 2020
Southampton UK
On the image shown on the attachment, a charger contains a coil wrapped around an iron core. The coil is plugged into the ac mains supply, and toothbrush also contains a cooil that sits around the iron core when the toothbrushed is placed on the charger to recharge the battery of the toothbrush.

Question: Describe how the charger is able to charge the low voltage battery.

I understand :

1.Mains supply V --> 2.allows for ac current to flow though coil --> 3. inducing an alternating magnetic fieldwithin the iron core --> 4. the change in flux through the secondary coil ( of the toothbrush) induces an secondaryy alternating emf in toothbrush coil (according to faradays Law)--> 5. therefore ac current in that circuit --> 6. diode is placed to prevent the a.c current discharging the battery in toothbrush

Looking at the diagram, I understand that this is a step down transformer, as there are visbinly less turns in the secondary coil compared to the first.

7. So there are less turns in secondary coil to induace a smaller secondary Emf and current than the first, suitable to charge a low voltage battery.

But what I do not understand is why the answer to point 4.

'It is a change in flux linkage '(Not Flux) 'that induces an alternating emf in the toothbrushes coil according to Faraday's Law' , and emphasises that it is the changing flux linkage by underlining it. But I do not understand why this is so? The reason I do not think the statement above is true is because, correct me if I am wrong :

through the whole tootbursh coil a magnetic flux of Φ , passes through the area of the coil, and it is this Φ value that changes due to an alternating magnetic field, whereas the value of N does not change, it is a constant in the case of the toothbrush coil as the number of turns are not changing. So what does it mean there is a change in flux linkage?

Does it mean a change of flux ΔΦ will induce an emf in one of the turns, but through N turns in series, a change of flux ΔΦ will induce an N*emf of one turn,
so flux linkage isnt a real thing like flux is of a magnetic field, rather it is value whose change over time, will tell you how much Emf is induced across a whole coil?

Or is it just the fact that E= - dNΦ /dt hence even if it were only the flux Φ changing, you would say change in flux linkage induces... not change in flux?

But if N is constant in this case, surely I can say it is only the change in flux (not flux linkage) as E= - N (dΦ /dt ).