Name it what you want. Its the meaning that counts.
It is common, because this grace is common to all of mankind. Rain falls on everyone. If rain fell only on the elect, the non-elect would always be coming to our farms to take our food. But God gives rain to all mankind. Its common.
God places a desire to worship in ALL of mankind. That is common grace.
God places the Law in ALL of mankind. That is common Grace.
God restrains sin. If you think this world is bad, if God removed his hand, man may not last a full year. We would all kill each other. This restraining by the hand of God is for all mankind. It is common grace.
Are all people born in a land that has a church or two or even 100 churches that they can walk to? Are all people born in a land that has Christian Radio? That alone is not common to all men, but it is grace given by God.
Of course I don’t agree with your “Determinist” view of election. And I’m already aware of the Calvinist view of common grace which holds to the reasoning of maintaining general morality. This is where I would think you should realize the intent in “your” definition. I’m am more interested in of whether or not your view the intent being a genuine offer of salvation for all who receive your brand of common grace; it logically does not. So the paraphrased quote I gave of, “Grace is no longer grace, if it does not include the saving intent of the Giver” is contingent on the word “saving” for the full meaning of intent. The Calvinist’ intent in common grace, as far as I know, does not believe it is sufficient for salvation. It believes there is a “special” grace for that that one. Seems you’re trying to have it both ways.
As for the old argument of whether the Gospel is offered to all men and how, I’ll not get started on that one, there are plenty of treads about this already. I will just state how I see it:
If a man received your brand of common grace with a genuine intent to save, and prayed to know God, would God not send him all (whatsoever) he needed regardless of where he was at or how many churches where around the corner? I believe He would; I believe that promise of intent is in His Word,