They can KNOW when they are truly saved, but can be deceived into thinking they are when they are not.
They may not know they are a false convert. Cameron is 100% right here. You asked if someone can " If someone comes forward and says they want to be saved. So we pray together, and they confess they are a sinner in need of Jesus as their Savior and they ask Him to come into their heart, you're saying they are not saved??
The answer is Yes! they can do all that and even think they are sincere when doling so, and NOT be saved. I did. I "felt it in my heart" at an alter call, went up front and repeated the prayer you're talking about, and was told to write down that date, I was saved. Even went a few weeks later to get baptized. I was not saved. So yes you can do what you said and NOT be saved. A prayer does not save us, our "decision" does not save us. ONLY God can save us.
That's why when I was saved, I did none of this. I was deceived into thinking I had already tried that "Christian thing", so when I hit my knees in the repentance God granted me I didn't recite a prayer, nor did I even call out for God or consider Him at all in this moment, when I hit my knees I knew and said 1 thing and 1 thing alone. "I can't do this anymore". I saw myself in all the glory we have in and of ourselves, I saw my self in truth, as powerless and pathetic as I truly was.
Look, I understand what you mean when you say what you're saying, and in theory what you say is sound. I'm not trying to act like what you are saying is wrong and you need to be corrected. I am just trying to share with you that doing that may be how God saves you, and I am in know way trying to suggest that we shouldn't always be trying to point others to the King and true reconciliation with our Creator. I am trying to point out that we can not guarantee people that once they say these certain words they are born again. That's just not true and we have to submit our way to Him completely, and this isn't a small easy thing to teach or do. Only His Spirit can do it and we have to be very mindful of that at all times.