In the same chapter of the Skandha Purana (Sankara Samhita Chapter 2) look at the verse no 33 and 34 which gives name of 4 Vaishnava Puranas (as per that chapter):
विष्णोर्हि वैष्णवं तद्वत् तथा भागवतं तथा।
नारदीय पुराणं च गारुडं वैष्णवं विदु:।। 34
Vishnu Purana, Bhagavata Purana, Narada Purana and
Garuda Purana; these (four) are known as Vaishnava Puranas.
After some verses the same chapter says that 10 Shaiva Puranas are for the Brahamanas and are Satvik Purana and these 4 Vaishnava Puranas are Tamasic and are for Kshatriyas:
दश चैव पुराणानि सात्विकानि विदुर्बुधा।
ब्रुवत्येव पुराणानि ब्राह्मणानां कथा: शुभा:।
दश चैव पुराणानि हिंसा दोष पराङ्मुखम्।
वैष्णवानि च चत्वारि तामसानि मुनीश्वराः।
क्षत्रियाणां श्रुता धर्मा: तेषु तद्देवता हरि:।। 49
These ten Shaiva Puranas are known to be as Satvik Puranas. These Puranas with auspicious stories are for the Brahamanas. These ten Shaiva Puranas are beyond doshas like Himsa. Know these Four Vaishnava Puranas as Tamasic. They are for Kshatriyas and Devata is Hari in them.
Various Puranas give their own classification of Satvik, Rajasik and Tamasic Puranas Like: