How to interpret "cessation"?

Firstly, there are the five aggregates and the five clinging aggregates. From SN 22.48 and from this question:

“And what, bhikkhus, are the five aggregates? Whatever kind of form there is, whether past, future, or present, internal or external, gross or subtle, inferior or superior, far or near: this is called the form aggregate. Whatever kind of feeling there is … this is called the feeling aggregate. Whatever kind of perception there is … this is called the perception aggregate. Whatever kind of volitional formations there are … these are called the volitional formations aggregate. Whatever kind of consciousness there is, whether past, future, or present, internal or external, gross or subtle, inferior or superior, far or near: this is called the consciousness aggregate. These, bhikkhus, are called the five aggregates.

“And what, bhikkhus, are the five aggregates subject to clinging? Whatever kind of form there is, whether past, future, or present … far or near, that is tainted, that can be clung to: this is called the form aggregate subject to clinging. Whatever kind of feeling there is … that is tainted, that can be clung to: this is called the feeling aggregate subject to clinging. Whatever kind of perception there is … that is tainted, that can be clung to: this is called the perception aggregate subject to clinging. Whatever kind of volitional formations there are … that are tainted, that can be clung to: these are called the volitional formations aggregate subject to clinging. Whatever kind of consciousness there is, whether past, future, or present, internal or external, gross or subtle, inferior or superior, far or near, that is tainted, that can be clung to: this is called the consciousness aggregate subject to clinging. These, bhikkhus, are called the five aggregates subject to clinging.”

Secondly, from this answer, you can see that the Buddha still had intentions and could plan his afternoon. So, living arahants still have sankhara of the volition kind. Living arahants also have sankhara of the kind where you have physical, verbal and mental activities (from this answer).

In SN 12.38, we read that intention and planning alone are insufficient for the continuation of suffering. There must also be latent tendencies (anusaya). This was discussed in this answer.

So, living arahants still have sankhara. What they do not have is latent tendencies (anusaya), defilements (kilesa), effluents (asava), fetters (samyojana), craving (tanha) and clinging (upadana).

More precisely, living arahants have the aggregate of sankhara but not the clinging-aggregate of sankhara, because they have stopped clinging. See SN 22.48.

Thirdly, according to Iti 44 (quoted below), once liberated, the living arahant's ignorance, suffering and burning of the aggregates with the fires of aversion, hatred and delusion have ended. The fires have been quenched, but the embers are still glowing. These embers are the five aggregates (not clinging-aggregates). They will continue till the arahant's physical death. This is explained in Thanissaro Bhikkhu's commentary.

And what is the Unbinding property with fuel remaining? There is the case where a monk is an arahant whose fermentations have ended, who has reached fulfillment, finished the task, laid down the burden, attained the true goal, ended the fetter of becoming, and is released through right gnosis. His five sense faculties still remain and, owing to their being intact, he is cognizant of the agreeable & the disagreeable, and is sensitive to pleasure & pain. His ending of passion, aversion, & delusion is termed the Unbinding property with fuel remaining.

And what is the Unbinding property with no fuel remaining? There is the case where a monk is an arahant whose fermentations have ended, who has reached fulfillment, finished the task, laid down the burden, attained the true goal, ended the fetter of becoming, and is released through right gnosis. For him, all that is sensed, being unrelished, will grow cold right here. This is termed the Unbinding property with no fuel remaining."

Commentary (Thanissaro):
With fuel remaining (sa-upadisesa) and with no fuel remaining (anupadisesa): The analogy here is to a fire. In the first case, the flames are out, but the embers are still glowing. In the second, the fire is so thoroughly out that the embers have grown cold. The "fuel" here is the five aggregates. While the arahant is still alive, he/she still experiences the five aggregates, but they do not burn with the fires of passion, aversion, or delusion. When the arahant passes away, there is no longer any experience of aggregates here or anywhere else.

Now, let's apply all of the above to the stock phrase that you had quoted.

"Now from the remainderless fading & cessation of that very ignorance comes the cessation of fabrications.

A foetus has ignorance. With that as the cause, the fabrications (physical, verbal and mental) arise. The rest of the clinging aggregates will also arise and dependent origination will become fully operational.

After he grows up, he becomes a living arahant. When ignorance has been overcome with no remainder, the five clinging aggregates will cease. But the five aggregates (of the non-clinging kind) remain till his physical death, as we saw in Iti 44.

From the cessation of fabrications comes the cessation of consciousness. From the cessation of consciousness comes the cessation of name-&-form. From the cessation of name-&-form comes the cessation of the six sense media. From the cessation of the six sense media comes the cessation of contact. From the cessation of contact comes the cessation of feeling.

Once again, this simply says that after liberation from suffering, the five clinging aggregates have ceased, because their cause has ceased. But not the five aggregates. The five aggregates, which are like glowing embers after the fire has ceased, will keep functioning till physical death.

Name-and-form is simply the mind-body link, which is anyway part of the five aggregates. The nidanas of fabrications (sankhara), consciousness (vinnana), name-and-form (namarupa), six sense media (salayatana) and feeling (vedana) are all located within the five aggregates. Hence, the distinguishing of aggregates from clinging aggregates in SN 22.48, is fully applicable to these nidanas. Also, the notion from Iti 44 that the five aggregates remain operating in an untainted way till physical death for a living arahant is also applicable to these nidanas.

From the cessation of feeling comes the cessation of craving. From the cessation of craving comes the cessation of clinging/sustenance. From the cessation of clinging/sustenance comes the cessation of becoming. From the cessation of becoming comes the cessation of birth. From the cessation of birth, then aging & death, sorrow, lamentation, pain, distress, & despair all cease. Such is the cessation of this entire mass of stress & suffering."

When ignorance has ceased, the clinging aggregates cease. With this, craving, clinging, becoming and the rest of suffering, all cease for a living arahant.

The only thing that remains are the aggregates, untainted, till the living arahant's physical death.

If the five aggregates are different parts of a guitar, then dependent origination simply explains how the different parts of the guitar work together to create music (see Lute Sutta). Music here refers to the self (atta), which is what becoming (bhava) and birth (jati) is about.

For a living arahant, the music has stopped, but the guitar remains. One day, even that will be destroyed. The five clinging aggregates is like a guitar vibrating with music, while the five aggregates not subject to clinging is like a silent peaceful guitar.

Although I said music, it's more like noise. When the noise stops, you have peace. That's the living arahant's mentally cognized peace of nibbana.