A first remark
You've already posted a question about a hadith were you were in doubt about imam at-Tirmidhi's statement Dhikr for which Allah forgives sins in the day and night, hasan or gharib (commentary)?. While investigating I found ibn Hajar commenting on a comment of imam an-Nawawi -apparently in his al-Adhkaar- as imam an-Nawawi there stated that the sanad of abu Dawod's version was good, which made ibn Hajar correct it saying he might have some reasons to do so, however ... and he counted issues in the hadith.
Here again an-Nawawi claimed that at-Tirmidhi declared this as hassan saheeh ghareeb in this version -see here in al-Adhkaar- (the translation in your provided image however deviates from this). However at-Tirmidhi actually didn't use the term sahih (sound) here at all.
It is unclear to me, why imam an-Nawawi said so. Sheikh al-Arnaout for example said that imam an-Nawawi's statement about Sa'id ibn al-Marzuban ibn Sa'ad al-Baqqal was considered weak by a majority of scholars and there's no such an agreement as the expression of an-Nawawi may indicate.
Hassan in the terminology of at-Tirmidhi
Note that imam at-Tirmidhi as stated in an earlier answer already wrote two sentences explaining what he means when qualifying a hadith as hassan:
قَالَ أَبُو عِيسَى وَمَا ذكرنَا فِي هَذَا الْكتاب حَدِيث حسن فَإِنَّمَا أردنَا بِهِ حسن إِسْنَاده عندنَا كل حَدِيث يرْوى لَا يكون فِي إِسْنَاده من يتهم بِالْكَذِبِ وَلَا يكون الحَدِيث شاذا ويروى من غير وَجه نَحْو ذَاك فَهُوَ عندنَا حَدِيث حسن
Abu 'Isa (at-Tirmidhi) said: As for what we mentioned in this book is a hassan (good) hadith we only mean with it that its has a good chain of transmission. With us this is every hadith that is narrated that does not have in its chain of transmission those who are accused of lying, and that the hadith is not irregular (shaadh) and that is narrated without a wording (or version) that is similar to that of yours.
In other words: Any hadith that
- has transmitters of an acceptable level of trust (not known to be fabricators, liars).
- And a wording of the hadith should not be irregular nor have a remarkable difference to the wording of a known accepted hadith.
could fall in this category.
Also note that hassan sahih in the terminology of imam at-Tirmidhi isn't an indicator that the hadith is of good level of authenticity. So it can hardly be regarded as hassan.
Other narrations of the hadith
The imams abu Dawod 1 & 2 (Both Sahih/Al-Albani), an-Nasa-i (Sahih/Darussalam same narrator chain as one of abu Dawod's versions) and ibn Majah 1 (Hassan/Darussalam) & 2 (Sahih/Darussalam)) reported versions of this hadith that are more reliable however they might have a difference in wording: prophet vs. messenger is a major difference in these.