And therein lies the bulk of their problems. Language can be precise when so employed. Imprecise language leads to imprecise thinking. Imprecise thinking leads to false understanding. Scripture is what it is and when theologians "help" God with better terminology, they essentially, IMO, erect a tower of Babel.
Scripture defines regeneration, and God had a purpose in its definition being what it is. To substitute a definition more to our liking is to substitute God's purpose.
As I've stated earlier I believe the term "regeneration" is poor and I think that now more than ever. Different people have different interpretation for "regeneration". From my understanding "regeneration" is not a single event but a series of events which takes place in a believers' life. I would lay regeneration out as the following events: 1) predestined, 2) the calling (illumination and confession), 3) justification; 4) glorification (Rom 8:30).
If your goal is to support a doctrine wherein 'regeneration' now includes calling (drawing) so as to support the argument that regneration precedes belief, then yes you will need a 'better' definition for regeneration.
Think carefully about how your desired finding is shaping your data.
My goal is to understand the scriptures-not to support a doctrine. I would include "drawing" as part of 1) illumination and 2) confession. As with Cornelius God illumination lead to Cornelius confession.
That is how the scriptures define drawing. Not as man making an intellectual choice. There is no scriptural support for that interpretation.