What determines the fate of a cell with respect to differentiation?
Cell differentiation, cell fate and cell mapping is an interplay of accessible evolutionary strategies/programmes and responses to dynamic environmental cues such as specialized hormones (e.g. morphogens) and physical parameters and constraints. That is putting it very broadly. It is a complex issue, if L. Wolpert's PLOS assays are any indication. I compiled a few links to get you started.

Specifically, reappraising the topic of your cited classical experiment are R.Moore et al:


  The classical cell sorting experiments undertaken by Townes and
  Holtfreter described the intrinsic propensity of dissociated embryonic
  cells to self-organize and reconcile into their original embryonic
  germ layers with characteristic histotypic positioning. Steinberg
  presented the differential adhesion hypothesis to explain these
  patterning phenomena.....


(Source: Robert Moore, Kathy Q. Cai, Diogo O. Escudero, Xiang-Xi Xu, Cell Adhesive Affinity Does Not Dictate Primitive Endoderm Segregation and Positioning During Murine Embryoid Body Formation, Genesis. 2009 September; 47(9): 579–589)

Regarding: ...without something taking the lead and actively coordinating...

On a hunch that you are not talking about complexity, attractors and polarity, I would like to share a few words. You may completely disregard them if you'd like.

My personal experience (from my country) is that few biologists delve into "that" branch of EvoDevo, because a profound understanding involves calculus, modeling and a knack for theoretical biology. Yet, afaik few struggled comprehending the evolutionary logic of making an organism based on Albert's book chapters alone.

See:


Reaction–diffusion system
Cell fate determination
Fate mapping
Asymmetric cell division
Pattern formation
Alberts Molecular Biology of the Cell describes this very well (can be accessed online for free)


I personally recommend G.Müller's authored/coauthored papers:

(open access)
http://homepage.univie.ac.at/gerhard.mueller/publications-papers.html

Disclaimer: I am not specialized in the field of EvoDevo or Theoretical Biology.
Views expressed here are my own.