In Shaivaism, the practice of meditation is recommended for liberation. What are the other ways to achieve liberation recommended by the other sects? I'm aware that there are different denominations in every sect. What are the methods of liberation they recommend and do they differ from the other denominations of the same sect. How and why?
A partial answer.
In Shaivaism, the practice of meditation is recommended for liberation.
Its true that Yoga is a proven way of achieving Moksha but what you say is not entirely true also.
Even in the so-called "Shaivism" the "Bhakti MArga" seems to be the best way to attain liberation.At least ,Shiva Himself says so in the Kurma Purana :
Parama Shiva asserted:
Naaham Tapobhirvividhairna Daanena na cheyjyayaa, Shakyo hi Purushair jaaturmutey Bhaktiamanuttamaam/
(I am always realizable by Bhakti or Pure Devotion and not necessarily by Tapasya, Daana or Charity and Yagnaas!)
Na madbhaktaa vinashyanti madbhaktaa veeta kalmashaah, Aadaavetat pratigjaanam na mey bhaktah pranashyati/ Patram Pushpam Phalam toyam madaaraadhana kaaranaat, Yo mey dadaati niyatah sa meyBhaktah Priyomatah/
My Bhaktaas would never be destroyed nor harmed but would be washed off their sins as I took a vow that my Bhaktas are protected. Bhaktas are merely required to perform my Puja with utmost sincerity and offer leaves, flowers, fruits and even water as I would be pleased).
So,as you can see,Parama Shiva himself is speaking so highly of the "Bhakti MArga".
Dhyanena maam prapashyanti kechirjjnanena chaaparey, Aparey bhakti yogena chaaparey/ Sarveshaameva bhaktaamaamishtah Priyataro mama, Yohi jnaanena maam nitya maadhaaraahyayati nyaayathaa/ Anye chaye trayo Bhaktaa madaaraadhana kaankshinah, Teypi maam praapunanteva naavartantey cha vai punah
Some persons seek my Darshan through Dhyaana or meditation, some by Jnaana, but those by Bhakti are most dear to me and they would have no re-birth
In all Vaishnava traditions it is said that the path of bhakti is recommended as the best. And in doing bhakti there are several practices, of which it is said that hari kirtana or chanting holy names of Lord Vishnu and glorifying Him is the best, and it is especially recommended for this age of Kali or Kali yuga. There are many statements in the scriptures about that, and here is one of those verses, it's Bhagavatam 12.3.52:
kṛte yad dhyāyato viṣṇuṁ
tretāyāṁ yajato makhaiḥ
kalau tad dhari-kīrtanāt
"Whatever result was obtained in Satya-yuga by meditating on Viṣṇu, in Tretā-yuga by performing sacrifices, and in Dvāpara-yuga by serving the Lord’s lotus feet can be obtained in Kali-yuga simply by hari-kīrtanāt or chanting the holy names of Lord Vishnu (Hare Kṛṣṇa mahā-mantra.)"
Further explanation and additional quotes you can see in the commentary at the above vedabase link.