There are very few Christian groups that stipulate what women coming to worship should wear, and what they should not wear. I have yet to come across one that 'lays down the law' on such personal matters, and certainly Peter was not trying to give a legalistic approach to this matter.
What quite a few groups ask of women coming to worship is that they dress modestly (as opposed to provocatively), and that they do not flaunt their wealth (if they are wealthy) via jewellery, fabrics, and such-like material things. In the New Testament all Christian women are urged not to be lewd in language, or lewd in clothing, or lewd in jewellery, or lewd in any way whatsoever. They are expected to be modest, and not flaunting anything, having, rather, a quiet and respectful manner that commends Christ to all others. That is the foundational principle involved here.
The question seeks to explore the meaning of one particular Greek word in one verse to get to grips with whether women may wear adornments and/or adorned clothes. Yes, the word in question clears some of that up. It is the word 'kosmos' generally translated 'world', as in God's created, ordered, beautiful world that we inhabit. Of interest is that in the Greek Septuagint version of the Hebrew Scriptures, the Hebrew word for 'ornament' is translated as 'kosmos' - world. (See, for example, Isaiah 49:18 - clothed with ornaments, as a bride; Jeremiah 4:30 - putting on scarlet [clothing], ornaments of gold, painting the eyes.)
The Old Testament speaks of brides adorning themselves, and of queen Esther coming in (uninvited) to her husband's throne-room wearing her crown (gold, and bejewelled no doubt) and her richest, most fabulous apparel. There is not a word of criticism about any of that, for such women were not coming into a mixed assembly to worship God. In 1 Peter 3:3 they are. That is the critical difference. Those gathering to worship God should never come to be looked at by others, to be admired, or to be considered of high rank. And that applies equally to the men.
However, it is women who are rather more likely to want to appear attractive and beautiful in public. If they use clothes, jewellery, or make-up to draw attention to themselves, they are violating Christian principles for people at worship.
This has got nothing to do with avoiding clothes with a pattern of flowers printed on them! There would be no such thing in first century common useage. What was noticeable were plain colours of scarlet, purple and royal-blue. Those colours denoted royalty, and common people not only could never afford such expensive cloth, in some parts of the world they would be put to death for daring to wear what was reserved for royalty. Yes, Lydia was a seller of purple; high-end commerce indeed. But although the Bible never says whether she ever wore clothes in that colour, what Peter said in the verse in question would apply equally to her as to all other Christian women. "If you've got it, don't flaunt it", is the New Testament principle.
Today, patterned cloth with flowers or tartan, or plain but in outstanding colours, is as common as muck. Nobody needs to be rich to have such clothing. Nobody would stand out in a congregation of worshippers wearing such cloth. But provocatively dressed people would, as would those dripping in jewellery (and I don't mean paste or plastic jewellery), or with wildly expensive coiffured hairstyles reminiscent of outlandish wigs in the Georgian court.
The world loves such things, and the more outlandish the appearance, the more it applauds because it is being entertained by all of that. That is the sense of 'worldliness' that should be avoided in gatherings meant to worship the holy God. There should be no place for entertainment, or worldliness, including that of the way the world loves to flaunt clothing and ornaments. And although Peter spoke of women, the principles he laid down apply equally (in this day and age) to men who might come into a gathering of worship to attract attention to themselves.