These are radically different propositions, yet I've seen discussion flit from one to the other as if they were conceptual neighbours. Type 1 collapse is a familiar historical event, and hence inherently plausible. Whether one wants to say that humankind experienced a "collapse" in the early 20th century is, however, questionable. "Collapse" typically means "a sudden, complete failure or breakdown". Using this term to describe a 30% drop in GDP that lasts for less than a decade seems hyperbolic.

I have several times asked that someone offer an argument that a  Type 2 collapse (or worse) could plausibly result from climate change that isn't far beyond the worst-case IPCC scenarios. Pointing to past civilisational collapses has been the best argument to date. But what were those collapses?

A recent book review in Science discusses the history of collapse and what has been said about it:

It is perhaps not too surprising that in a time of widespread anxiety about global environmental change, the collapse of civilizations is a topic of intense interest. For example, building on scholarly work in archaeology from the last 20 years, Jared Diamond's recent bestseller, Collapse (1), merged an apocalyptic vision of environmental degradation with an upbeat lesson from the self-help literature, suggesting that societies have chosen to succeed or fail. Spectacular failures, real or imagined, certainly have broad popular appeal: Regional abandonments of settlements in the U.S. Southwest are typically represented as "mysterious disappearances" even though such shifts represented common and effective strategies for managing ecological challenges--and, of course, even though Pueblo peoples are still very much present in the region today.

I've seen this example offered here, and also the next:

Similarly, the Maya are famous primarily for having a complex political and social order involving monumental architecture, an order that failed spectacularly in the Terminal Classic period. Although the Classic Maya collapse involved both political change and large-scale depopulation, even there life went on: as Diane Chase and Arlen Chase describe in their contribution to After Collapse, Post-classic Mayan society restored Classic-period institutions of symbolic egalitarianism and shared rule while rejecting Terminal Classic strategies that more clearly marked personal inequalities. Neither Mayan civilization nor Mayan peoples disappeared, a long-term record of continuity that seems to be the norm rather than the exception, as the articles in this volume make clear.

What is meant by "collapse" in the study of civilisations?

Here it is worth clarifying what contributors to this volume mean by collapse. As Schwartz enumerates, collapse "entails some or all of the following: the fragmentation of states into smaller political entities; the partial abandonment or complete desertion of urban centers, along with the loss or depletion of their centralizing functions; the breakdown of regional economic systems; and the failure of civilizational ideologies." Note that this definition refers only to the collapse of complex political structures and that death and destruction are conspicuously absent. Although the focus of After Collapse is decidedly on continuity and renewal, archaeological studies of collapse itself have always recognized that civilizational traditions and peoples rarely disappear.

Collapse in this sense would be a disaster of historic proportions, but far short of the apocalyptic visions that the word often conjures up.

If I were to continue from here, I'd point out the ways in which modern civilisation is made more adaptable by its technologies, less dependent on local resources by its transportation networks, and less culturally fragile by its distribution across multiple continents and societies. I have great difficulty imagining a climate-change scenario in which disruption reaches the level of Type 2 collapse. Disruption of ecosystems, resource flows, and so on, could be enormous, disastrous, and deadly, yet fall far short of driving every society on the planet into a pre-industrial state.

Global civilisation contains tough and resourceful people, diverse cultures, and loosely coupled systems. Would spill-over from local societal collapses bring down the rest? I think that the weakness of the victims and the potential brutality of the rest gives a strong indication of the answer.