He thought that making up a missed fast was like a voluntary fast and that it was permissible to break it

Making up missed Ramadaan fasts is obligatory, and it is not permissible for a person to break such a fast unless there is a legitimate shar’i reason. If a person starts to make up a missed fast, then he must complete it. He is not like one who is observing a naafil fast, because in that case he is in charge of the matter; if he wishes he may break his fast and if he wishes he may continue it. 

See question no. 49985. 

It was narrated that Umm Hani’ (may Allaah be pleased with her) said: “O Messenger of Allaah, I broke my fast when I was fasting.” He said to her: “Were you making up anything?” She said: No.” He said: “Then it does not matter if you were observing a voluntary fast.” Narrated by Abu Dawood, 2456. Classed as saheeh by al-Albaani. This indicates that it would matter if she broke her fast when observing an obligatory fast. What is meant by “it would matter” here is that it would be a sin. 

With regard to what happened between you, the expiation for intercourse is only required in the case of intercourse during the day in Ramadaan itself. Based on this you do not have to do anything, and all she is required to do is to make that day up again. And you (both) have to repent to Allaah and resolve not to do such a thing again. 

Ibn Rushd said: The majority are agreed that no expiation is required for deliberately breaking a fast observed to make up a missed Ramadaan fast, because it does not come under the same category with regard to sanctity of time, i.e., Ramadaan. 

Bidaayat al-Mujtahid, 2/80 

It says in Fataawa al-Lajnah al-Daa’imah, 10/352: Expiation is only required of one who has intercourse during the day in Ramadaan because of the sanctity of that time. With regard to a fast observed to make up for a missed fast, no expiation is required, according to the more sound of the two scholarly opinions.