The concept of logical implication or material conditional — its names are legion — is ever a source of logical bedevilment. It has proved no different in the discussion of Peirce’s logical graphs however much potential they bear for resolving the confusion if given the chance.
Here’s a link to an article on Logical Implication I originally wrote for Wikipedia where I attempted to synthesize the more coherent perspectives on the subject I had learned from various communities of usage over the years.
The last time I looked at the Wikipedia relic it had devolved into the usual run-of-the-mill obscurities of nominalism so let’s not go there. The Wikiversity edition still looks okay.
I’ve been meaning for years now to add the graphical representations — maybe these recent discussions will nudge me to do that …