The Madhyamika tradition, as I understand it, focuses on nirvāṇa as the cessation of ignorance. So do the early texts, actually. It’s primarily just these discussions that keep focusing on the nirvāṇa after death. 
As I see it, that focus of ours is in large part due to the idea (held by some) that there is some kind of ultimate reality or consciousness that is nirvāṇa, which is experienced in life and after death. Those kind of notions of continuity automatically lead us to discuss the situation after death. In my view of nirvāṇa, not only is that wrong, but it isn’t all that practical either. I would be happy if we could focus more on nirvāṇa while alive. The ending of craving and ignorance is also continually indicated to be the ultimate goal in the suttas, not some situation after death (whatever it may be), which is interesting to reflect on in and of itself.
But here we are. 
That said, the Madhyamika tradition does make the very same twofold distinction. (Edit: I’m wrong. see yeshe’s point below.) From Candrakīrti’s Prassanapada (519), which is a commentary on the MMK, and very influentual in the Tibetan tradition. In Sprung’s translation:
Nirvāṇa has been described by the illustrious one as twofold for those persons who lead a chaste life, who are practising the discipline leading to perfect realization, and who are committed to living according to the Truth: namely, a Nirvāṇa with a residual base and a Nirvāṇa without a residual base.
On the one hand, Nirvāṇa with residual base (sopadhiśeṣa) is conceived as resulting from the total extinction of the basic afflictions—ignorance, desire and the rest. […] Nirvāṇa with a residual base means continuing to exist conjoined with a residual base. What kind of Nirvāṇa is this? It consists of nothing more than the bare factors of personal existence freed from the deceptive afflictions such as the belief in a substantial personal self; it is like a town from which all criminal gangs have been purged. That is Nirvāṇa with a residual base.
In the second case, the Nirvāṇa in which there are not even the bare factors of personal existence [skandhas] is Nirvāṇa without a residual base (nirupadhiṣeśa). Where the residual base has vanished, it is comparable to a town from which all criminal gangs have been purged and which is itself destroyed. In this connection it is said, ‘The body has collapsed, ideas and perceptions gone. All feeling vanished, all dispositions quiescent and consciousness itself non-existent.’ […]
So it is that Nirvāṇa without residual base is attained by the cessation (nirodha) of the factors of personal existence.
From Siderits’ comment on the MMK:
For nirvāṇa is said to be of two types, with and without remainder. The former involves abandonment of the defilements, so that cessation of rebirth is assured but still involves psychophysical elements resulting from past karma, so one is still embodied. The latter comes about when one’s karma is exhausted, so that the causal series of psychophysical elements is destroyed. Both involve cessation. The former involves the cessation of false views of an existing “I,” while the latter involves cessation of the psychophysical elements.
I’ve seen people been confused about the concept of “without remainder” when this specifically refers to the defilements. In that case it does not mean no remainder of anything whatsoever. The Pali texts use such terms also; for example in MN18:
[…] the end of the underlying tendencies to desire, repulsion, views, doubt, conceit, the desire to be reborn, and ignorance. […] This is where these bad, unskillful qualities cease without anything left over [of those unskillful qualities themselves!]
The term for ‘without anything left over’ is aparisesa. This is not the nirupadhiṣeśa nirvāṇa that Candrakīrti refers to above, which he says happens when “the body has collapsed”.
It may well be true that in extant Tibetan circles this twofold distinction is no longer made. That I am not aware of.