Lord Ganesh's name 'Vinaayak' in the above context means 'Garud', 'vi (bird) naayak (foremost) or 'vihang naayak' - who is the Naayak (leader) of Vihag (birds), that is Garud.

There is a commentary on Ganapati Sahastranama by Bhaaskar Raaya Makhin with the name "Khadyot". Among the 1000 names of Ganapati, there is a name called "Vinaayak" and "Sapt Maatrikaa Sevitah". Bhaaskararaya Makhin interprets the name Vinaayak from the standpoint of Vishnu and says that vi means Garud as well as Jeev. So, it is also explained as the one who is alighted on Garud as well as controller of Jeev who are also called Bhoot. And the name Sapt Maatrikaa Sevitah implies to 7 Mothers as Braahmee, Maaheshwaree....

There are certain temples in Aandhra that were constructed by Badaamee Chaalukya where Vinaayak is seen along with Sapt Maatrikaa and sometimes along with Veerbhadra. This can be seen in Aalampur, Jogulambaa Temple which is in Kurnool District of Aandhra Pradesh which is the 5th Shakti Peeth.

Now, the references of Vinaayak, Sapt Maatrikaa and Chatur-bhaginee are traced back to a lost Tantra which is Taamasik in nature called "Vinaashik Tantra" where Vinaayak is projected as having negative character who creates obstacles to the Upaasak. There is a Prayog in this Tantra that is deployed to counter the attack of Vinaayak, Sapt Maatrikaa and Chatur-bhaginee.

The name Vinaashik stuck because among one of the Moorti of Shiv, he has a Shikhaa (tuft) of hair and in the tuft, a Veenaa is seen to be hanging down. Now, the Baahya Upaasanaa is of two types depending upon the Varn and Aashram - (a) Vaidik and (b) Taantrik. The Vaidik Upaasanaa is for three Varn (Braahman, Kshatriya, and Vaishya) and the Taantrik Upaasanaa is for Shoodra.

So, basically, Shankar urges us to follow Vaidik Maarg and worship the Devataa like Aaditya (Shankaraananda Bharati mentions in His Bhagvad Geetaa commentary for Shlok 9.25) and eventually inculcates Ahangrahopaasanaa which paves the way for Chitta Shuddhi (cleansing of mind and heart). So, Dev Vrataa indicate Devayaan Maarg and Pitri Vrataa indicate Pitri Yaan Maarg.