Here is where i have a problem with reformed theology on this matter. You say drawing with the HS but leave out scripture. Youre not alone every reformed person i have talked to does it. Its not the HS that does the drawing it is the gospel.
You have correctly noticed that there are various degrees of "reformed". The drawing of the HS affects the person by conviction and by him realizing that the gospel is true and that he is in jeopardy if he doesn't come to Christ. The means to do this IS God's word, from preaching or reading the Bible.
Now, I assume you are some form of regular Baptist or fundamental Baptist. I was too. You know we pray for people to get saved. What are we doing? We are asking God to intervene in their lives either by convincing them or by arranging circumstances to help them get saved. We are assuming that this intervention is necessary or the person will not be saved. In that case the grace or intervention was essential, was it not? Something happened that would not have happened had the HS not acted.
Now here's the differences. The non-reformed Baptist believes that this grace can be resisted. I believe as I explained above that by definition if it can be resisted then it was not the intervention you prayed for. Reformed people do believe that the grace can be resisted - but not in the ones who get saved - or they wouldn't have gotten saved. For those folks, the grace was overcoming.
Now, if you "decide" to believe as I have described above, do you not see how you can say your faith was a gift? You wouldn't have believed without the HS. Otherwise, why pray for folks - they're gonna have to decide on their own.
I believe in addition, that there are enough verses that show that the work of the HS is MORE than conviction and convincing. I believe there is an enlightening or regeneration (being born again) that really does occur. And I believe it was planned by God. I know that a lot of non-reformed people believe that you come by faith and then get born again. I believe it's the other way around.
Like I said earlier, "Reformed" or Calvinist takes in a wide range of belief. Some are so into the facts of God's sovereignty that they forget that even if salvation is all of God - the elect person is a rational being and there has to be an intelligent interaction between the two parties or it has no meaning. You actually have to believe. Some even say you are justified before being created if you are elect. Other like Spurgeon, Bonar, a group of Puritans called the Marrow men, describe salvation in much more interactive terms that you as a Baptist would be familiar with.