Evangelicals love to ask "Have you accepted Jesus as your personal Lord and Savior?" but most people are unaware of the damning contradiction behind this question that is tearing Evangelicalism apart daily. The contradiction, or better yet self-condemning dilemma, is summed up in what is known as the Lordship Salvation controversy. The concept of Lordship Salvation teaches that Jesus is not just Savior, but Savior and Lord. This is specifically understood to mean that Jesus doesn't just save you, but He's also your master whom you must obey. This means that anyone living a life of sin cannot be truly a believer, since anyone who has "saving faith" will prove this by obeying Jesus, principally by turning away from sin and producing good fruit. Indeed, there are plenty of texts that would suggest this very thing (e.g. 1 John 2:4; Mat 7:15-23). To Catholic ears, this sounds perfectly reasonable. So what's the big deal?
The "big deal" is that those advocating the concept of Lordship Salvation are those in the Reformed (Calvinist) camp who were combating the charge of anti-nomianism (literally: anti [moral] law). From the start of the Pretend Reformation, Catholics have pointed out that the Reformed doctrines of Faith Alone and "Once Saved, Always Saved" will naturally cause Christians to let their guard down and fall into sin. Many will (logically) think that if they cannot lose their salvation, then they're free to not follow the moral law (and thus be anti moral law). These antinomians are seen as "Christians" who effectively confess Jesus as their Savior but not as their Lord to obey. In response to this charge, the first Calvinists came up with an ad hoc solution: they said that anyone who truly has "saving faith" is guaranteed to produce good fruit and avoid sin, and all others who fail to do this will be exposed as deceived liars who were "never saved in the first place." This is how they explained how a person was justified by faith alone while maintaining the Bible's warnings about the need to produce good works and avoid sin. (They often say "Faith alone saves, but true saving faith is never alone, it's always accompanied by good works.") But there's a fatal flaw in this "solution," and Catholics and many non-Calvinist Protestants recognized it.
The fatal flaw in the concept of Lordship Salvation was that not only did the Bible never teach good works are guaranteed, this doctrine also had the negative effect of causing despair whenever a Christian sinned (especially since Christians regularly sin!). For example, if good works are guaranteed, then what happens when a Christian falls into sin? Were they never saved in the first place? How many good works must a Christian see before they can be confident they were really saved? What about those Christians who turn to lives of sin later on in life? This line of questioning put the Calvinist in a bind, for the Calvinist has no good answer for this dilemma, since Calvinists on one hand admit true Christians struggle daily with sin (and their salvation cannot be lost), while on the other hand admit good works are guaranteed and lack of which demonstrates one was never originally saved. They cannot give any coherent or concrete measurement for if someone is truly saved, and this 'examining oneself' has the effect of undermining any notion of looking to Jesus and being assured of your salvation. In fact Calvinists would have to conclude that sinners like Kind David who lived with unrepentant murder and adultery for a year were never saved (even though Calvinists believe David was saved and never lost his salvation). In fact, the (Calvinist) Westminster Confession reluctantly admits this embarrassing contradiction in their theology: "Nevertheless, they [true believers] may, through the temptations of Satan and of the world, the prevalency of corruption remaining in them, and the neglect of the means of their preservation, fall into grievous sins; and, for a time, continue therein..." (Ch XVII:III) So lack of good works, including remaining in serious sin for a time, does not prove anything, since true believers can have this happen to them. Thus, Lordship Salvation was seen (quite logically) as a dangerous heresy against the comforting Gospel message of assurance and self-contradiction by many non-Calvinist Protestants, particularly many modern Evangelicals.
This is why many Evangelicals today will call Calvinism a heresy that destroys a person's assurance, making a mockery of the Gospel, and rightly so, since in the Lordship Salvation scheme a person cannot really have any assurance. Instead, they will (if they're honest with them self, though most wont publicly admit it) realize that as soon as they fall into sin or stop producing good works then they must question whether they were really saved in the first place. This mentality was especially engrained in the Puritan mind, which is why they were so "rigid" in condemning anything that even potentially could be sinful and striving non-stop to engage in good works, since this was (logically) the only way they could convince themselves they had true faith. So the irony of ironies is that Protestantism was invented to give an "assurance" which Catholicism allegedly did not give (which is not accurate), and in the end it resulted in a system that made people more uncertain and more dependent on their works than ever before. The options the systems provide are plain: either one can have assurance from just believing, or one must do good works, and more good works, (and more good works, and more good works, and persevere in good works, and persevere further in good works,) before they can "know" their original faith in Jesus that happened 37 years ago was genuine "true saving faith" and thus (finally! maybe!) have "assurance".
All that said, the "conservative" Evangelicals in the anti-Lordship crowd will make it clear that even though one does not need to do good works as a Christian, since their salvation doesn't depend on that, that Christians should still do good works out of thanksgiving to God and to make a good testimony to their neighbor. So these "conservatives" do not advocate sinning, but they are also very clear that a Christian could sin, even backslide into a life of sin, and still be saved. This is precisely where the modern notion of "The Sinner's Prayer" came from, where they reason (quite logically) that all one really has to do is pray a short prayer along the lines of "Jesus, I don't deserve eternal life, but I believe you died for my sins, and so I accept you as my savior and ask you to come into my heart today." And with that, they're saved. This is where they then "give their testimony," and when they retell the story include the fond memory of "that day" 23 years ago when they "got saved" and didn't have to worry if they couldn't live a perfect or even mediocre Christian life.
The problem with that is it goes right back to the original dilemma the Pretend Reformers were in, that of stopping the (logical) slide into antinomianism, which is why many Protestants today will (logically) go to such extremes as saying without shame that anyone who confesses Jesus as their Savior is saved and can live as they please. The result of this fully-realized, logically consistent, outworking of Faith Alone is that our society today is full of "Christians" living unashamedly sinful and scandalous lives, all the while thinking they're safe and sound before God. Those in the Reformed camp (rightly) viciously criticize this, calling it "cheap grace" and "easy believism," but their ad hoc solution (described earlier) doesn't get them out of this dilemma.
The only way out of that vicious self-condemning dilemma is to abandon Justification by Faith Alone. As a Catholic response to the original question about accepting Jesus as Lord and Savior, I'd respond by saying something to the effect that, yes I accept Jesus as Lord and Savior, and by that I mean that He saved me so that I can faithfully obey Him as Lord, and thus hopefully be judged worthy to enter Heaven by cooperating with His grace. When the Reformed Protestant is caught off guard and objects, I'd ask them what accepting Jesus "as Lord" means to them, and proceed to point out the dilemma they're in. Those espousing the Sinner's Prayer are a lot harder to reason with, since they're typically not interested in systematic theology and will brush off any of the warning passages of Scripture (e.g. Galatians 5:19-21; 1 John 3:10). The good news is that more and more people are seeing Sola Fide for the dangerous and unChristian doctrine that it is. Unfortunately, so much damage has and will continue to be done by it for the foreseeable future.