What is the correct term to use, Shaivism or Shivaism? I could not find a good source about how the word Shaiva (Saiva, Caiva) appeared and when.
From Wikipedia of Shaivism,
Shaivism ([Śaivam]) (Devanagari: शैव संप्रदाय) (Tamil: சைவம்) is one of the major traditions within Hinduism that reveres Shiva as the Supreme Being or its metaphysical concept of Brahman. The followers of Shaivism are called "Shaivites" or "Saivites"
Devotees of Shiva are Shaiva
Devotees of Vishnu are Vaishnava
Hence Shaiva is the correct pronunciation.
Shaivism / Śaivism is the correct English term to describe the worshippers of god Śiva.
However, note that, the 'ism' is an English concept.
In the Sanskrit language, the original word for Śaivism sect is 'Śaiva (शैव)' or 'Śaivā (शैवा)'.
Quoting from the Nārada Mahā-Purāṇa
Verses 56-57, Chapter 82, Uttarabhāga:
शिवं शैवा वदेत्येनं प्रधानं सांख्यवेदिनः ।।
योगिनः पुरुषं विप्राः कर्म मीमांसका जनाः ।। ८२-५६ ।।
The followers of Shiva (Śaivā) call that Supreme Being Śiva,...
As regards Shivaism, I don't think such words in Sanskrit exist. But, from the English POV sure, one can identify it that way, like Vishnuism, but that's not the Sanskrit way of referring.
It took some digging to get it answered:
Śaivism is formed from śaiva as Brāhmanism is from brāhmaṇa. Grammatical explanation of what is a Śaiva according to Aparāditya’s Yājñavalkyasmŗți-tīkā: Śaiva is one who studies or who knows the śaivam, which is in turn defined as the scriptural system propounded by Śiva (śivaproktaṃ śāstram). Formation: The use of the aṆ suffix (i.e., taddhita formation) can mean “taught by him” (tena proktam), giving us the word śaivam. Then, I can understand that śaivam ≠ śaivism.