It’s obvious by the last sentence of the picture that the author is trying to link together Luke’s mention of the first use of the word ‘Christian’, the fact that it happened in Antioch, the fact that Ignatius was bishop of Antioch, and that Ignatius used the word ‘Catholic’ to say … that Catholics are Christians?
Let’s look at Acts 11:26-28, shall we?
…and when he had found him, he brought him to Antioch. For a whole year they met with the church and taught a great many people. And in Antioch the disciples were first called Christians.Claudius reigned from AD 41 to AD 54 (see here), so this had to have occurred during that time-period. Luke wrote Acts some time before AD 62, the year the apostle Paul was martyred in Rome (since Luke doesn’t mention it in Acts). Ignatius wrote his epistle to the Smyrneans sometime before his martyrdom in AD 110, so his use of the word ‘Catholic’ occurred some 50 to 60 years after the word ‘Christian’ was first used – and you can pretty much bet the word ‘Christian’ was in use well before Ignatius’ time.
Now in these days prophets came down from Jerusalem to Antioch. And one of them named Agabus stood up and foretold by the Spirit that there would be a great famine over all the world (this took place in the days of Claudius). (emphasis mine)
Ignatius did use the word ‘Catholic’ (‘Big-C Catholic’), but he wasn’t using it in the sense we do now – as the Catholic Church. He was using it in the ‘small-c’ manner, meaning ‘universal’.
If you go to Ignatius’ Epistle to the Smyrneans (here) and scroll down to Section 97, you read the following:
 'The bishop, argues Ignatius, is the centre of each individual Church, as Jesus Christ is the centre of the universal Church.' — LIGHTFOOT. This is the earliest occurrence in Christian literature of the phrase 'the Catholic Church' (η καθολικη εκκλησια). The original sense of the word is 'universal.' Thus Justin Martyr (Dial. 82) speaks of the 'universal or general resurrection,' using the words η καθολικη αναστασις. Similarly here the Church universal is contrasted with the particular Church of Smyrna. Ignatius means by the Catholic Church 'the aggregate of all the Christian congregations ' (Swete, Apostles' Creed, p. 76). So too the letter of the Church of Smyrna is addressed 'to all the congregations of the Holy Catholic Church in every place.' And this primitive sense of 'universal' the word has never lost, although in the latter part of the second century it began to receive the secondary sense of 'orthodox' as opposed to 'heretical.' Thus it is used in an early Canon of Scripture, the Muratorian fragment (circa 190-210 A.D.), which refers to certain heretical writings as 'not received in the Catholic Church.' So too Cyril of Jerusalem, in the fourth century, says that the Church is called Catholic not only 'because it is spread throughout the world,' but also 'because it teaches completely and without defect all the doctrines which ought to come to the knowledge of men.' This secondary sense arose out of the original meaning because Catholics claimed to teach the whole truth, and to represent the whole Church, while heresy arose out of the exaggeration of some one truth and was essentially partial and local. The use of the word in this passage by Ignatius has been urged as an indication of the late date of the epistles. But the fact that it is used in its primary sense is on the contrary an indication of early date. (emphases mine)It's also of note that the Apostle’s Creed (ca. AD 700), the Nicene Creed (ca. AD 325/381), and the Athanasian Creed (ca. AD 500) all use the word ‘catholic’ to mean ‘universal’, and, as you can see, each was written well after Ignatius wrote his epistle. So Ignatius’ use of the ‘capital-C-Catholic’ doesn’t mean the ‘Catholic Church’, it means the ‘universal church’.
As to whether or not Catholics are Christian, here’s a suggestion: read God’s Word – the Bible – and then read the Roman Catholic Church’s word – the Catechism of the Catholic Church; compare and contrast the two, and see their similarities and their glaring differences.
That ought to answer the question much better than the picture up at the top of this post.