Alhamdulillah wasolatu wasalam ‘ala rosulillah.

This issue is somehow technical, may Allah grant us understanding.
Basically, we have 3 assumptions on how to complete the missed rakat(s);
1. ‘Itimam’ (to complete): meaning that whatever thelatecomer catches up with his imam in a solat is regarded as the beginning ofhis solat and what he prays on his own is the end of his prayer. He started itwith Imam and completes it on his own. The evidence quoted is that the Prophet(peace and blessings of Allaah be upon him): “…..whatever you catch up with,pray, and whatever you have missed, complete it.” (Bukhari and Muslim)
2. ‘Qadah’ (to make up): meaning what the latecomer catchesup with his Imam is the latter part of his solat and what he makes up is thefirst part of his solat. The evidence quoted is that the Prophet (peace andblessings of Allaah be upon him), “…..whatever you catch up with, pray, andwhatever you have missed, make it up.” (Musim)
3. ‘Jam’u’ (reconciling the itimam and qadah and acceptingthem all): using both aforementioned assumptions together at the same time.(This is very common)
Starting from the last assumption, the issue of jam’uusually comes up where two accepted hadiths controverts each other. The first and second hadith seems to controvert each other but they are not. Scholars differ on the meaning of the end part of the second hadith ‘….fa aqiduu’ from the word ‘qadah’.
Where there is a difference in translation like this in the sharia, the best place to confirm the actual meaning is in the Quran or Hadith.
In the Quran, there are 10 different usages of ‘Qadah’ (see Zahru kimam ‘ala qisoti yuusuf ‘alayhi salaam) we have ‘itimam’ (to complete)(or nearest in meaning of ‘to complete’) as one of the meaning and we don’thave “to make it up”. E.g “When you have finished [qudiyat] As-Salaah (the congregational prayer)” Suratu n Nisa’ 4:103 etc
So, both hadiths have the same meaning which established thefirst assumption. Therefore, only the first assumption (Itimam) is correct because it has clear cut evidence.
People will still tell you “but according to Malikis or Shafi’s….” this statement and the like should not disturb you for these reasons;
1. Maliks said is different from Imam Maliki said.
2. “Imam….. Said” what is the evidence of Imam….on that assumption.
3. All our Imams (Imam Hanafi…) warned us not to followtheir words until we know the evidence presented to establish the assumption.(See Sifatu solat …By Imam Albani, Sahih fiqhu Sunnah by Abu Malik)
4. There is no aspect of religion that doesn’t have evidence, this is solat confirm it yourself please!
To this end, I am not the first person to make this clarification, See Imam Albani view on this issue and others. (Ref: Masailu‘ilimiyya wafatawah shar’iyyah limam Muhammad Nasirudeen al Albaniy Fil madinatwal imarat Compiled by ‘Abdu Mun’im Saleem etc).
Insha Allah, I will give examples on how to complete the missed rakat(s) of each solat (5 daily solat) starting from Solatul Fajr.

SOLATUL FAJR
If you miss the first rakat and join Imam from the second rakat, that second rakat of Imam is your first rakat. After Tesleem of Imam youwill stand up observes your second rakat as you normally observe second rakahof solatul fajr (Alfatiah + other Surah loudly, ruku’.., sujud.., sitting,tashaud and tesleem).

SOLATU ZUHUR
If you miss the….
FIRST RAKAT
It means that you caught up 3 rakats with Imam. The 3 rakats you caught up with Imam is therefore your first rakat, second rakat and thirdrakat, remaining only fourth rakat. After Tesleem of Imam you will stand upobserves your fourth rakat as you normally observe fourth rakah of solatul Zuhur (Al-fatiah only, ruku’.., sujud.., sitting, tashaud and tesleem).
FIRST AND SECOND RAKATS
It means that you caught up 2 rakats with Imam. The 2 rakats you caught up with Imam is therefore your first rakat and second rakat remaining third and fourth rakat. After Tesleem of Imam you will stand upobserves your third rakat as you normally observe third rakah of solatul Zuhur (Alfatiah only, ruku’.., sujud.., and stand up for fourth rakat. You will observe fourth rakat as you normally observe fourth rakah of solatul Zuhur(Alfatiah only, ruku’.., sujud.., sitting, tashaud and tesleem).
FIRST, SECOND AND THIRD RAKATS
It means you caught 1 rakat with Imam. The 1 rakat you caught up with Imam is therefore your first rakat remaining second, third and fourth rakats. After Tesleem of Imam you will stand up observes your second rakat as you normally observe second rakah of solatul Zuhur (Al-fatiah + other surah silently, ruku’.., sujud.., sitting, tashaud and stand up for third rakat). Observe the third rakat as you normally observe third rakah of solatul-Zuhur (Al-fatiah only, ruku’.., sujud.., and stand up for fourth rakat. You will observe fourth rakat as you normally observe fourth rakah of solatul Zuhur(Alfatiah only, ruku’.., sujud.., sitting, tashaud and tesleem).

SOLATU ‘ASR
If you miss the….
FIRST RAKAT
It means that you caught up 3 rakats with Imam. The 3 rakats you caught up with Imam is therefore your first rakat, second rakat and third rakat, remaining only fourth rakat. After Tesleem of Imam you will stand up observes your fourth rakat as you normally observe the fourth rakat of solatul‘Asr (Al-fatiah only, ruku’.., sujud.., sitting, tashaud and tesleem).
FIRST AND SECOND RAKATS
It means that you caught up 2 rakats with Imam. The 2 rakatsyou caught up with Imam is therefore your first rakat and second rakat remaining third and fourth rakat. After Tesleem of Imam you will stand up observes your third rakat as you normally observe third rakah of solatul ‘Asr (Al-fatiah only, ruku’.., sujud.., and stand up for the fourth rakat. You will observe fourth rakat as you normally observe forth rakah of solatul ‘Asr(Alfatiah only, ruku’.., sujud.., sitting, tashaud and tesleem).
FIRST, SECOND AND THIRD RAKATS
It means you caught 1 rakat with Imam. The 1 rakat you caught up with Imam is therefore your first rakat remaining second, third and fourth rakats. After Tesleem of Imam you will stand up observes your second rakat as you normally observe second rakah of solatul ‘Asr (Al-fatiah + other surah silently, ruku’.., sujud.., sitting, tashaud and stand up for the third rakat). Observe the third rakat as you normally observe third rakah of solatul‘Asr (Al-fatiah only, ruku’.., sujud.., and stand up for the fourth rakat. You will observe fourth rakat as you normally observe fourth rakah of solatul ‘Asr(Al-fatiah only, ruku’.., sujud.., sitting, tashaud and tesleem).

SOLATU MAGRIB
If you miss the….
FIRST RAKAT
It means that you caught up 2 rakats with Imam. The 2 rakats you caught up with Imam is therefore your first rakat and second rakat remaining only third rakat. After Tesleem of Imam you will stand up observes your third rakat as you normally observe the third rakah of solatul magrib(Alfatiah only & silently, ruku’.., sujud.., sitting, tashaud and tesleem).
FIRST AND SECOND RAKATS
It means that you caught up 1 rakat with Imam. The 1 rakat you caught up with Imam is therefore your first rakat remaining second and third rakat. After Tesleem of Imam you will stand up, observes your second rakat as you normally observe second rakat of solatul ‘Magrib (Al-fatiah + Surah loudly, ruku’.., sujud.., sitting, tashaud and stand up for the third rakat.You will observe third rakat as you normally observe third rakat of solatul Magrib (Al-fatiah only & silently, ruku’.., sujud.., sitting, tashaud and tesleem).

SOLATU ‘ISHAI
If you miss the….
FIRST RAKAT
It means that you caught up 3 rakats with Imam. The 3 rakats you caught up with Imam is therefore your first rakat, second rakat and third rakat, remaining only fourth rakat. After Tesleem of Imam you will stand up observes your fourth rakat as you normally observe the fourth rakah of solatul‘Ishai (Al-fatiah only & silently, ruku’.., sujud.., sitting, tashaud and tesleem).

FIRST AND SECOND RAKATS
It means that you caught up 2 rakats with Imam. The 2 rakats you caught up with Imam is therefore your first rakat and second rakat remaining third and fourth rakat. After Tesleem of Imam you will stand up observes your third rakat as you normally observe third rakah of solatul ‘Ishai(Al-fatiah only silently, ruku’.., sujud.., and stand up for the fourth rakat.You will observe the fourth rakat as you normally observe fourth rakah of solatul ‘Ishai (Al-fatiah only & silently, ruku’.., sujud.., sitting,tashaud and tesleem).

FIRST, SECOND AND THIRD RAKATS
It means you caught 1 rakat with Imam. The 1 rakat you caughtup with Imam is therefore your first rakat remaining second, third and fourthrakats. After Tesleem of Imam you will stand up observes your second rakat asyou normally observe second rakah of solatul ‘Ishai (Alfatiah + other surahloudly, ruku’.., sujud.., sitting, tashaud and stand up for the third rakat).Observe the third rakat as you normally observe third rakah of solatul ‘Ishai(Alfatiah only, ruku’.., sujud.., and stand up for the fourth rakat. You will observe fourth rakat as you normally observe fourth rakah of solatul ‘Ishai(Alfatiah only & silently, ruku’..., sujud.., sitting, tashaud and tesleem).

MOD'S SHOULD DO THE NEEDFUL I BELIEVE THIS WOULD EDUCATE/CORRECT THOSE OF US THAT OBSERVE OUR SOLAT WRONGLY..
May Allah accept our Ibadaat.