Is it true that a women who wears perfume or makeup in public should take ghusul when she comes home?
The hadith you are asking about is an authentic hadith in Sunan an-Nasā'i:
عَنْ أَبِي هُرَيْرَةَ قَالَ قَالَ رَسُولُ اللَّهِ صلى الله عليه وسلم: إِذَا خَرَجَتِ الْمَرْأَةُ إِلَى الْمَسْجِدِ فَلْتَغْتَسِلْ مِنَ الطِّيبِ كَمَا تَغْتَسِلُ مِنَ الْجَنَابَةِ
It was narrated that Abu Hurairah said: "The Messenger of Allah [SAW] said: 'If a woman goes out to the Masjid, let her perform Ghusl to remove perfume as she would perform Ghusl to remove Janabah (impurity following sexual activity).'" (This is an abridged form of it.)
In (Arabic: مرقاة المفاتيح شرح مشكاة المصابيح) Vol. 3, pp. 837, Al-Qāri said in his commentary about the hadith above that when a woman who is wearing perfume is going out, she should perform full ghusl (cover the entire body with water) to remove any traces of perfume. If the perfume is only in a specific spot of her body, she may wash only that specific spot. If the perfume is on her clothes, she should change her clothes.
In the commentary on the hadith above in Fayd al-Qādīr (Arabic: فيض القدير), Vol. 1, pp. 334, Al-Manāwi said that if a woman goes out to the masjid or otherwise, if she is wearing perfume, she should perform ghusl as a recommendation (nadb, Arabic: ندب), not as an obligation (wujūb, Arabic: وجوب).
There is another hadith graded hasan (some scholars elevate it to sahih) that shows the level of prohibition of a woman going out with perfume as if the woman is committing adultery (this is a literary device to show that it is strongly prohibited):
عَنْ أَبِي مُوسَى عَنِ النَّبِيِّ صلى الله عليه وسلم قَالَ: كُلُّ عَيْنٍ زَانِيَةٌ وَالْمَرْأَةُ إِذَا اسْتَعْطَرَتْ فَمَرَّتْ بِالْمَجْلِسِ فَهِيَ كَذَا وَكَذَا يَعْنِي زَانِيَةً
Narrated Abu Musa that the Prophet (ﷺ) said: "Every eye commits adultery, and when the woman uses perfume and she passes by a gathering, then she is like this and that.'" Meaning an adulteress.
In another hadith in Sahih Muslim, the Prophet ﷺ explicitly prohibits women from wearing perfume when attending the masjid:
عَنْ زَيْنَبَ امْرَأَةِ عَبْدِ اللَّهِ قَالَتْ قَالَ لَنَا رَسُولُ اللَّهِ صلى الله عليه وسلم: إِذَا شَهِدَتْ إِحْدَاكُنَّ الْمَسْجِدَ فَلاَ تَمَسَّ طِيبًا
Zainab, the wife of Abdullah ibn 'Umar, reported: "The Messenger of Allah (ﷺ) said to us: 'When any one of you comes to the mosque, she should not apply perfume.'"
As one should take one's adornment at every masjid (Qur'an 7:31), and in light of the hadith above in Sahih Muslim prohibiting women from wearing perfume when going out to the masjid, it can be assumed that wearing perfume when going out for other purposes is also prohibited.
Perfume was widely used in the muslim world since the early times of Islam. Muslims like Jābir ibn Hayyān, Al-Kindi and Avicenna contributed significantly to the development of Western perfumery and it eventually arrived to Europe with the crusaders.
We know that the Prophet Muhammad (pbuh) used perfume and that he didn't reject perfume when it was given to him as a gift:
It was narrated that Aishah said: "I put perfume on the Messenger of Allah for his exiting Ihram, and I put perfume on him for his Ihram, perfume which was not like this perfume of yours" she meant that it does not last. (Sunan an-Nasa'i 2688)
Narrated 'Azra bin Thabit Al-Ansari: When I went to Thumama bin `Abdullah, he gave me some perfume and said that Anas would not reject the gifts of perfume. Anas said: The Prophet (ﷺ) used not to reject the gifts of perfume. (Sahih al-Bukhari 2582)
Now to your question: The Prophet Muhammad (pbuh) ordered every muslim to perform Ghusl on friday and to wear perfume after the Ghusl (if he can find it).
Al-Bara bin Azib narrated that: Allah's Messenger said: "It is a duty for the Muslims, that they perform Ghusl on Friday, and that each of them wear some of his family's perfume. If he does not find any, then water is a perfume for him."
So we can say that perfume is seen as a good thing in Islam. So there shouldn't be a reason for a woman (or a man) to take perform Ghusl after coming home if perfume was used before.
Also, there is a small limitation for using perfume.
It was narrated that Anas Said: "The Prophet forbade men to perfume themselves with saffron. (Sunan an-Nasa'i 2706)