Hadiths were collected after a delay, so their words were lost due to the distance from the Prophetic era and the passage of time. Those who collected hadiths did not write the hadith with the exact words but rather with their meanings. The words belonged to the narrators themselves, so a hadith is essentially the sayings of these men who are not free from errors.
First Point
The mentioned claim is not correct because hadiths are narrated with both words and meanings, as the first narrators of hadith were the Companions and the Followers, whom Allah, the Lord of Glory, has praised:
والسابقون الأولون من المهاجرين والأنصار والذين اتبعوهم بإحسان رضي الله عنهم ورضوا عنه وأعد لهم جنات تجري تحتها الأنهار خالدين فيها أبدا ذلك الفوز العظيم
Reference: (At-Tawbah: 100)
Translation: And the first forerunners among the Muhajireen and the Ansar and those who followed them with good conduct—Allah is pleased with them and they are pleased with Him. He has prepared for them gardens beneath which rivers flow, wherein they will abide forever. That is the great attainment.
Because changing words necessarily changes the meaning, can it be expected from the Companions that they attribute to the Prophet ﷺ what he did not say, while they heard from the Prophet ﷺ:
من كذب علي متعمداً فليتبوأ مقعده من النار
Reference: (Sahih Bukhari: 1291)
Translation: Whoever deliberately lies upon me, let him prepare his place in Hell.
Second Point
There is instruction and encouragement from the Prophet ﷺ himself to be careful with words, as mentioned in the hadith:
وبنبيك الذي أرسلت، فإن مت من ليلتك فأنت على الفطرة، واجعلهن آخر ما تتكلم به) قال البراء: فرددتها على النبي ﷺ، فلما بلغت: آمنت بكتابك الذي أنزلت” قلت: ورسولك الذي أرسلت، فقال النبي ﷺ :لا، ونبيك الذي أرسلت
Reference: (Sahih Bukhari: 247)
Translation: …The Prophet you sent, I have believed in him. So if he dies that very night, he will die upon the natural disposition (fitrah), and recite this supplication at the end of all matters. Bara’ رضي الله عنه says that I recited this supplication again in front of the Messenger of Allah ﷺ. When I reached (اللهم آمنت بكتابك الذي أنزلت), I said the word (ورسولك). The Prophet ﷺ said: No! Say it like this (ونبيك الذي أرسلت).
Similarly, the hadith praises the one who conveys the hadith exactly as he heard it:
(…فأدّاها كما سمعها…)
Reference: (Sunan Tirmidhi: 2657)
Translation: Delivered exactly as it is to others.
Third Point
As far as mentioning the hadith with meaning and interpretation is concerned, it is permissible when necessary and under specific conditions, some of the important conditions are:
➊ The one narrating the hadith should be well-versed in the Arabic language, have complete mastery over the meanings and concepts of the words, and be aware of the intentions of the speaker of the statement he is narrating.
➋ There arises a need to explain the hadith with its meaning, and at that time the narrator does not fully remember the exact words.
➌ The hadith should not consist of adhkar (remembrances) and supplications whose exact words have been prescribed by the Shariah, because some words are used in the worship of Allah Almighty, and there is absolutely no room for alteration in them.
Fourth Point
A large part of the Prophet's ﷺ instructions is transmitted with the exact words of the Prophet. For example: the words of the Adhan and Iqamah, the words of Azkar and Duas, and the Hadith Qudsi. Besides these, a significant portion of the Ahadith related to rulings and ethics consists of Fa'li Ahadith (i.e., the blessed actions of the Prophet ﷺ) or Taqreeri narrations. The meaning of "Taqreer" is that some action was performed in front of the Prophet ﷺ and he did not object to it, i.e., Ahadith about various matters and events that occurred with or in the presence of the Prophet ﷺ.
The words of such narrations are anyway from the companion of the Prophet or the narrator, who reports from their observation, and the original words do not exist to which adherence can be required.