Yes, it is correct. From responsibilities of the parents against their children is to choose good names for them. Each name is actually an adjective and someone who is repeatedly called by an adjective would accept effects from his/her name. Specifically naming the children by the name of the Ahlul-Bayt peace be upon him has been highly encouraged, if this naming is for the sake of their love.
Among the good names that the parents can choose for their children only a few of the names of Allah has been banned, as they are adjectives of Allah and no other one, maybe Mutakkabir being among them if I am not wrong.
Rasul can be a very good name or not, depending on the perspective of the parent with regard to this name. Rasul implies someone who has been given a Risalah (mission), and Risalah doesn't need to be only prophecy. Look e.g. in Quran when Allah calls three person Rasul whie they were not any prophet:
وَاضْرِبْ لَهُم مَّثَلًا أَصْحَابَ الْقَرْيَةِ إِذْ جَاءَهَا
الْمُرْسَلُونَ / إِذْ أَرْسَلْنَا إِلَيْهِمُ اثْنَيْنِ
فَكَذَّبُوهُمَا فَعَزَّزْنَا بِثَالِثٍ فَقَالُوا إِنَّا إِلَيْكُم
مُّرْسَلُونَ
Set forth to them, by way of a parable, the (story of) the Companions
of the City. Behold!, there came messengers to it. / When We (first)
sent to them two messengers, they rejected them: But We strengthened
them with a third: they said, "Truly, we have been sent on a mission
to you." {YaSin, 13,14}
And we know those three person were from Hawariyun of Jesus peace be upon him and they were not prophet. Another example is the crow that Allah sent to Qabil to show him how he should hide his brother's body in the ground, in this case the crow was also a Rasul as it has had a mission:
فَبَعَثَ اللَّـهُ غُرَابًا يَبْحَثُ فِي الْأَرْضِ لِيُرِيَهُ كَيْفَ
يُوَارِي سَوْءَةَ أَخِيهِ ۚ قَالَ يَا وَيْلَتَىٰ أَعَجَزْتُ أَنْ
أَكُونَ مِثْلَ هَـٰذَا الْغُرَابِ فَأُوَارِيَ سَوْءَةَ أَخِي ۖ
فَأَصْبَحَ مِنَ النَّادِمِينَ
Then Allah sent a raven, who scratched the ground, to show him how to
hide the shame of his brother. "Woe is me!" said he; "Was I not even
able to be as this raven, and to hide the shame of my brother?" then
he became full of regrets- [Al-Ma`idah, 31]
Note that بعث is also usually used for prophets, when Allah choose them as His prophet and give them missions, but this is used also for the resurrection day and etc.
And examples are more than to be said here, like the successor of every prophet had also a mission and could be considered as Rasul. In other words, we have two kinds of Rasul, oin one case it has a general meaning and in another case it is specifically stated about the prophets.
About Nabi you should be able to find similar justification, but it largely depends on the culture of the society, in some cultures Nabi is used only for prophets but in other cultures as you have mentioned many might choose this adjective as the name of their children and of course, they do not mean that their children would become a prophet.
Godspeed