Personally, I would not equate this with 'Convicting Grace' but 'Prevening Grace'. Here they are again...
1.) Prevenient Grace. The saving work of God begins not by our being attentive to prevening grace, but by grace that attends us and awakens our attentiveness. The focus is not first of all upon our cooperation initiative by which we imagine ourselves coming early to God, pleading to cooperate. Rather, the initiative comes from grace prevening prior to our first awakening to the mercy and holiness of God.
Prevenient grace elicits "the first wish to please God, the first dawn of light concerning his will, and the first slight transient conviction of having sinned against Him." Grace works ahead of us to draw us toward faith, to begin its work in us. Even the first fragile intuition of conviction of sin, the first intimation of our need for God, is the work of preparing, prevening grace, which draws us gradually toward wishing to please God. Grace is working quietly at the point of our desiring, bringing us in time to despair over own righteousness, challenging our perverse dispositions, so that our distorted wills cease gradually to resist the gifts of God.
Grace works antecedently to conversion to convict freedom of its falleness, and its need for a radical reversal, repentance, a reversal that is only possible in view of God's justifying grace, which meets us on the cross, of which we in time may become aware. At each stage we are called to receive and respond to the grace being incrementally given. Prevenient grace does not justify, but readies for justification, giving us the desire for faith, which is the one condition of justification.
The chief function of prevenient grace is to bring the person to a state of grace. Prevenient grace is that grace that goes before us to prepare us for more grace, the grace that makes it possible for persons to take the frist steps toward saving grace.
2.) Convicting Grace. Prevening grace leads toward convicting grace, which begins not with our self-initiated determination to repent but by the grace that elicits a determination to repent. Prevenient grace brings one to the point of attentiveness to one's own sinfulness, asking for works meet for repentance. that does not mean that works evidencing repentance are justifying works, since no work justifies, but that the threshold of grace is being entered by penitence.
Do you see the difference?