Someone told me that, in order to get a residence permit, you need to fill out a form for the issue/ replacement of the residence card. Is this the correct form?

No. If you would like to receive a residence card, you need to apply for temporary residence permit. The form for the replacement of the residence card can be submitted  when the data contained in your current residence card have changed. If your current  residence card that is about to expire, and you would like to prolong your stay in Poland, you should apply for another residence permit.