The connection of 'Raghunath' with Raam

Background on Raghunath

According to legend, Raghu was one of the great renowned King of Ikshvaku dynasty. According to the Vishnu Purana (4.4), Raam was the great grandson of Raghu: 

The son of Khatwánga was Dírghabáhu; his son was Raghu; his son was Aja; his son was Daśaratha. The god from whose navel the lotus springs became fourfold, as the four sons of Daśaratha, Ráma, Lakshmańa, Bharata, and Śatrughna, for the protection of the world. Ráma, whilst yet a boy, accompanied Viswámitra, to protect his sacrifice, and slew Tádaká.

Raghu itself means "fast one" and legend goes that it is derived from Raghu's chariot riding abilities. (More: https://www.wisdomlib.org/definition/raghu)

The Sanskrit word nātha (नाथ) literally means "lord, protector, master." The related Sanskrit term Adi Natha means first or original Lord, and is a synonym for Shiva, the founder of the Nāthas. Nath is a theological term found in all the dharmic religions that utilize Sanskrit. It is found in Vaishnavism (e.g. Gopinath, Jagannath), Buddhism (e.g. Minanath), and in Jainism (Adinatha, Parsvanatha).

So Raghu-nath means Lord of the Raghus, meaning Lord of the Dynasty of Raghu. Just like Lord Krishna is also called Yadunath meaning Lord of Yadavas, Lord Rama is called Raghunath meaning Lord of descendant of Raghus or the Lord of Ikshvaku dynasty.

What do I think of when I sing Raghunath

Earlier today I was hearing a debate between two people who were fervently fighting over what Raam means. One said it means the one that is ramaa hua or imbued within everyone, and the other one said it stands for Raam, the son of Dashrath as described above.  You can't say either one is completely right or completely wrong. I think both definitions are incomplete. In fact, any definition would be incomplete. Raam is not definable; you cannot put a box around Raam even if the box is infinite.  Tulsidas, who spent his life writing the story of Raam, the son of Dashrath describes this himself: he says Raam is "atarkya" ... cannot be understood by the mind, heart or words.

राम अतर्क्य बुद्धि मन बानी। मत हमार अस सुनहि सयानी॥
तदपि संत मुनि बेद पुराना। जस कछु कहहिं स्वमति अनुमाना॥

They all describe him according to their limited mind. Bhagat Tulsidas here sounds just like Guru Gobind Singh says:

Aap Aapni Budh Hai Jeti
Barnat Bhinn Bhin Tohe Teti.

I am glad that I don't define and translate words like "Raam" on my videos. It is because these words mean different things to people, and that is the beauty of it.

When I sing Raghunath in the shabad Tek Ek Raghunath I am remembering the one Ekonkaar (the primal sound energy) is called by many by different names. The one that is is within everyone and around everywhere. The one that is called by many different names but cannot be defined in words. But I also include all the stories and philosophies that I know about Raam; and I understand that I don't know.   

What I do know is that when I sing I do touch the essence of something that I cannot describe in words.  Raghunath touches that essence. In my experience, Raam, Krishna, Allah, Karima and all other names of the primal sound energy can all be used to touch the essence. The purpose of the singer is to sing, not to worry about which name it is. For instance Guru Nanak says, I love you through all your names: 

Balhaari Jaoon Jete Tere Nao Hain