What is the difference between Nirvana with residue and non-abiding Nirvana? Both these states apply to a Buddha while still alive and come to an end when the physical life of a Buddha comes to an end.
In reality there's no difference. There's only one Nirvana, not two, not three. The difference between the concepts is didactic:
the Nirvana with residue (or with fuel) is a Pali Canon concept defined as the extinction of attachment, aversion, and delusion. Even though the Buddha explicitly said that an arahant dwelling in this Nirvana still experiences agreeable and disagreeable as well as pleasure and pain, some students assume Nirvana must be a fixed sterile state, sort of like a perfect diamond with no flaws.
To counteract this delusion the Mahayana teachers coined a new concept of non-abiding (or non-localized) Nirvana. This Nirvana is explicitly defined as a non-fixed state of fully realized Shunyata or Emptiness, leaving no room for doubt or misunderstanding.
In both cases, we are talking about the same exact unconditioned Nirvana that Buddha taught. The non-abiding Nirvana is presented as a distinct (and superior) attainment to help the student let go of the idea of a fixed state and cultivate the atammayata.
In the pali sutta, nirvana in general refers to destruction of taints, in particular the removal of greed, anger & delusion. It translates as 'extinguishment'.
There are qualified and there is a definitive designation for this term.
"Bhikkhus, there are these two Nibbana-elements. What are the two? The Nibbana-element with residue left and the Nibbana-element with no residue left.
"What, bhikkhus, is the Nibbana-element with residue left? Here a bhikkhu is an arahant, one whose taints are destroyed, the holy life fulfilled, who has done what had to be done, laid down the burden, attained the goal, destroyed the fetters of being, completely released through final knowledge. However, his five sense faculties remain unimpaired, by which he still experiences what is agreeable and disagreeable and feels pleasure and pain. It is the extinction of attachment, hate, and delusion in him that is called the Nibbana-element with residue left.
"Now what, bhikkhus, is the Nibbana-element with no residue left? Here a bhikkhu is an arahant... completely released through final knowledge. For him, here in this very life, all that is experienced, not being delighted in, will be extinguished. That, bhikkhus, is called the Nibbana-element with no residue left.
"These, bhikkhus, are the two Nibbana-elements."
These two Nibbana-elements were made known By the Seeing One, stable and unattached: One is the element seen here and now With residue, but with the cord of being destroyed; The other, having no residue for the future, Is that wherein all modes of being utterly cease.
Having understood the unconditioned state, Released in mind with the cord of being destroyed, They have attained to the Dhamma-essence. Delighting in the destruction (of craving), Those stable ones have abandoned all being.
“Venerable sir, it is said, ‘the removal of lust, the removal of hatred, the removal of delusion.’ Of what now, venerable sir, is this the designation?”
“This, bhikkhu, is a designation for the element of Nibbāna: the removal of lust, the removal of hatred, the removal of delusion. The destruction of the taints is spoken of in that way.”
When this was said, that bhikkhu said to the Blessed One: “Venerable sir, it is said, ‘the Deathless, the Deathless.’ What now, venerable sir, is the Deathless? What is the path leading to the Deathless?”
“The destruction of lust, the destruction of hatred, the destruction of delusion: this is called the Deathless. This Noble Eightfold Path is the path leading to the Deathless; that is, right view … right concentration.”
Reverend, they speak of ‘extinguishment in the present life’. In what way did the Buddha speak of extinguishment in the present life?”
“First, take a mendicant who, quite secluded from sensual pleasures … enters and remains in the first absorption. To this extent the Buddha spoke of extinguishment in the present life in a qualified sense. …
Furthermore, take a mendicant who, going totally beyond the dimension of neither perception nor non-perception, enters and remains in the cessation of perception and feeling. And, having seen with wisdom, their defilements come to an end. To this extent the Buddha spoke of extinguishment in the present life in a definitive sense.”
There are certain 'meditative' attainments that are associated with this seeing with wisdom, seeing with wisdom is seeing with dhamma-eye, it sees the four noble truths in course of entering into & something from attainments associated with a cessation of perception & feeling, this seeing is what inadvertently destroys taints.
There are, monks, three unskilled ways of thought: thoughts of lust, thoughts of ill-will, thoughts of hurting. And these three unskilled states disappear utterly in him whose heart is well established in the four foundations of mindfulness, or who practices concentration on the signless.
"Indeed, monks, this concentration on the signless is greatly to be commended; the concentration on the signless, if developed and frequently practiced, is of great fruit, of great profit.
The signless, undirected, emptiness releases are based on the same principle which is the truth of cessation and are called different due to how they are first apprehended & approached. This is the theravadin interpretation.