Read what Paul wrote in 1 Cor 15:
1 Corinthians 15:
1 Moreover, brethren, I declare unto you the gospel which I preached unto you, which also ye have received, and wherein ye stand;
2 By which also ye are saved, if ye keep in memory what I preached unto you, unless ye have believed in vain.
3 For I delivered unto you first of all that which I also received, how that Christ died for our sins according to the scriptures;
4 And that he was buried, and that he rose again the third day according to the scriptures:
No read what Paul wrote in Rom 10:
Romans 10:
9 That if thou shalt confess with thy mouth the Lord Jesus, and shalt believe in thine heart that God hath raised him from the dead, thou shalt be saved.
10 For with the heart man believeth unto righteousness; and with the mouth confession is made unto salvation.
11 For the scripture saith, Whosoever believeth on him shall not be ashamed.
12 For there is no difference between the Jew and the Greek: for the same Lord over all is rich unto all that call upon him.
13 For whosoever shall call upon the name of the Lord shall be saved.
I am not sure that just "believing Jesus took the punishment for our sins by dying on the cross" is "how we're saved".
There is also the matter of confessing that Jesus Christ is Lord.
Is it possible for someone to believe Jesus took the punishment for their sins [everybody wants the savior] but that person doesn't really believe Jesus is Lord [not everybody wants the Lord]?
I believe we must follow the Lord Jesus Christ as Lord of our lives. Yes, He died for my sins. Yes, He rose from the dead. Yes, He ascended up into heaven and is seated at the right hand of God and He makes intercession for us (Rom 8:34). The Lord Jesus Christ is set at God's own right hand in the heavenlies far above all principality, and power, and might, and dominion, and every name that is named, not only in this world, but also in that which is to come (Ephesians 1:20,21).
Can a person believe that Jesus took the punishment for our sins without really believing that Jesus Christ is Lord? I don't know. But if a person does not acknowledge Jesus Christ is Lord, he or she is not saved (according to Rom 10).
Does this make sense to you?