An organization can execute in both a structural ambidextrous mode while simultaneously executing contextual ambidexterity.
Organisational Ambidexterity speaks to the organization being able to execute well on its day-to-day operations while keeping a steadfast eye on the future and emerging opportunities so as to secure the long-run viability of the entity. Some authors use the dichotomy of (a) being aligned and being adaptable at the same time or (b) embodying both tactical and strategic actions (c) exploring and exploiting simultaneously.
Organisations that focus on executing operations well and ignores future-scoping will enjoy near-term performance gains but will be eaten up by those rivals that had also invested in strengthening their future revenue streams through R&D, strengthening talent capacities, IP development and protection, new product and services development etc. On the other hand, those firms that only only on future opportunities but pay no attention to executing efficiently will not realize those futures as current revenues are appropriated by current day rivals (inefficiencies leading to low quality, poor prices, sub-standard marketing, customer defections, loss of market share).
High performance organizations are ambidextrous.
The older form structural ambidexterity speaks to a physical (structural) separation between those units in the organization that are exploring future opportunities for the next emergent profitable revenue streams of the future from those units that are dedicated to executing efficient production systems for competitive outcomes to the now market. Contextual ambidexterity, a newer concept, pushes ambidexterity decisions down to the individuals in firms (managers, employees) who are empowered to make these explore/exploit decisions for the firm. There are pros and cons to the prior concept of structural ambidexterity. Contextual ambidexterity on the other hand has a better fit with a knowledge-based economy and contemporary knowledge-worker talent-focused realities
The question is then , can both structural and contextual ambidexterity exist at the same time in the same firm ?
Yes they can. Individuals within the structural units can execute these align/adapt decisions and behaviors. In fact some authors (Julian Birkinshaw, Christine Gibson) argue that they complement each other although they recommended that structural separation not persist beyond a temporary state.
It is true that high performance organizations are ambidextrous.
However, almost every single researcher on this topic has a different definition of ambidexterity.
As you note, some have called it structural ambidexterity, others contextual ambidexterity, and a whole slew of others.
Gibson and Birkinshaw's (2004) paper was wonderful, but it is just one study and defined the exploration/exploitation concept as contextual.
The 'contextual' part was not well defined.
From research I have done on U.S. hospitals, the successful ones are ambidextrous. I published a book on this subject.
I prefer to avoid what I consider slight variations in terminology (contextual, structural, whatever) and just go by organization ambidexterity which I define as:
My assumption was (and is) that org amidexterity (OA) is a unitary paradoxical construct consisting of exploration and exploitation.
My definition of OA: :"The simultaneous pursuit of exploration and exploitation, within or between individuals and business units at the micro level and/or within or between organizations (intra/inter) at the macro level."
Think of this as a yin/yang effect.
These other descriptions of org ambidexterity are interesting to those in the Academy, but in practice whether something is termed contextual or strucutural; this has no meaning at all for active professionals.
As to your question, of course explore/exploit, align/adapt, etc - whatever you call it, of course it has to be done simultaneously as there are poor outcomes for doing just one or doing just another.
That's the notion of a unitary theory of org ambidexterity.
In my study of U.S. hospitals, I can say that hospitals are practicing organization ambidexterity, and they have to because of health reform efforts and declining revenues.