My analysis in a nutshell is that core activities are those that increase the sustainable competitive advantage of a company. Core activities create value for customers in a way that is hard forcompetitors to replicate, and by doing so increase the market power of the company. Investors notice this, and reward the company with a higher stock price.
Of course in today's market, core doesn't stay core for very long as competitors copy successful companies. At one point a web site to distribute marketing information was a core activity. Now it is a context activity, something that is required by the market that does not differentiate. Political factors also drive context to encroach on core. Everyone wants to feel important, meaning to feel like core, even though their activities might more reasonably be considered context. In most organizations, context activities compete for resources with core, and when they win, the company loses.
My recommendation is that companies never lose site of the distinction between core and context as they do business. Invest as much as possible in core activities. Seek to reduce costs and outsource context activities. If you have to cut spending in downturn, don't do it across the board, cutting core and context by equal measures. Instead, seek to actually increase your investment in core while making even more drastic cuts in context to achieve the total cost-reduction goal.