Whose is the 'grace of God that bringeth salvation' ? (Titus 2:11) This question is in the context of Protestant Trinitarianism.
There are two schools of thought within Protestant Trinitarianism : one, founded mainly in the Westminster Confession (or Savoy Declaration) which accepts that 'the Law is a rule of life for the Christian' and looks to the commandments (whether an expansion of ten, or whether two or whether one) as the 'Law' by which the Christian is to live.
This school of thought can be seen set forth in the Westminster Confession
The other school of thought accepts that salvation is by grace and that the Christian is 'freed from law' (that is to say, not just 'ceremonial law' but all law). The 'rule of the Spirit' is just that, they say - a governance of the Holy Spirit who brings the things of Christ and ministers them to the soul, whereby the believer receives the 'shedding abroad of the love of God' and is kept, in Spirit, walking in holiness by means of faith in Christ.
This school of thought can be seen set forth in the pages of The Gospel Standard.
I am interested in what the former school of thought teach, regarding Titus 2: 11-14 :

For the grace of God that bringeth salvation hath appeared to all men, (12) Teaching us that, denying ungodliness and worldly lusts, we should live soberly, righteously, and godly, in this present world; (13) Looking for that blessed hope, and the glorious appearing of the great God and our Saviour Jesus Christ; (14) Who gave himself for us, that he might redeem us from all iniquity, and purify unto himself a peculiar people, zealous of good works.

There is no mention in these few verses of the Law being administered in order to achieve either holiness or preparation for the last day. The latter school of thought hold that the 'grace of God' refers to the Holy Spirit and that his ministrations result in sanctification.
What, I would like to know, is the teaching of those who suggest that Christians are 'under the Law' (and the Westminster Confession states that this is as 'advice') in regard to these verses. Whose, exactly, is the 'grace' referred to ? And why, would they say, is Law completely missing from these exhortatory verses ?
I don't think there is such a distinction between the two camps here as you suggest. I think both of the documents you link to would agree that the Law expresses the will of God. The section of the Westminster Confession you quote about the rule of life starts out: 'Although true believers be not under the law, as a covenant of works'. I don't see that as particularly different from what the Gospel Standard say: &quot;We believe that the believer’s rule of life is the gospel, not the law of works given to Moses&quot;.
They are both saying that we are not under the Law in the same way that people of Israel were under the Law. They both agree that, as Jesus taught, we need a deeper obedience to the Law if we are to fulfil God's will for us.
So let me try to answer your specific question with respect to Titus 2:11-14. I come from a traditional Reformed (Anglican) perspective, this is how I understand the passage.
I believe that the grace of God is referring to Jesus and his work for us, rather than the Holy Spirit. Which would fit with 'appearing' - the grace of God 'has appeared' (v11) and then we are waiting for Jesus' 'appearing' (v13). Paul's classic definition of grace is Ephesians 2:1-10.
The grace of God teaches us to say no to 'ungodliness and wordly passions'. But the question is, what does Paul mean when he talks about ungodliness? How do we know what is godly and ungodly? &quot;I would not have known what sin was had it not been for the law&quot; (Romans 7:7). The Law taught him what ungodliness was - and the reason is because in our sinful states we need some guidance as to right and wrong. If there were no Law, we could deceive ourselves that we were OK.
So the Law reveals sinfulness. But it is not the means by which we attain godliness. This is a big message of Titus: we only attain righteousness by grace. Paul is here talking to a church who have been corrupted by those whose focus is the Mosaic Law (which I assume is what he means by &quot;the circumcision party&quot; in 1:10). This is why Paul does not refer to the Law in Titus 3. But he is not saying that it's an insignificant thing for Christians, but rather we can only achieve the righteousness God requires through grace.
I've just checked Calvin's commentary on this passage, and this is what he says:

“Piety” is religion towards God. “Righteousness” has place among men. He who is endowed with both of these lacks nothing for perfect virtue; and, indeed, in the law of God there is absolute perfection, to which nothing whatever can be added. But as the exercises of godliness may be regarded as appendages to the first table, so “temperance,” which Paul mentions in this passage, aims at nothing else than keeping the law, and, as I said before about patience, is added to the former as a seasoning.

So I think Calvin's view is not that we need to keep the Law as in by our own efforts, but the Law of God is perfection - and the only way we can keep it is by the grace of God.
Earlier this year I tried to express my view about Grace in a sermon on Titus 3. You might also find interesting the other sermons on the rest of Titus, linked from that page.