LiNgapurAna 70.170,173 says that they were created by brahmA, not Shiva. I have not been able to find references to Shiva incarnating as them in the text of this purAna, and it would be very helpful if you could provide a quote.
agre sasarja vai brahmA mAnasAnAtmanaH samAn,
RRibhuH sanatkumArash cha dvAvetAvUrdhvaretasau...
sanandaM sanakaM Chaiva vidvAMsaM Cha sanAtanam
BrahmA created the mental sons equal to himself. Among them Sanat and Rbhu were sages of matriculated sexuality... BrahmA created Sananda, Sanaka and SanAtana
In fact, the bhAgavata purAna (4.22.6) says that they are "bhavAgrajAn", "elder brothers of Shiva". This same purAna (1.3.6, 3.12.4, 3.17.15, 4.8.1, 10.87) calls them sons of brahmA many times. This is also confirmed in viShNu 3.14.12. I will not quote all of these sources here because of the amount of text.
The current text of the Shiva purAna contradicts itself plainly on who the kumAras are born from.
3.4.21-22 is a speech of Shiva that says:
tatrApi mama te putrAshchatvAro yogasAdhakAH,
bhaviShyanti mahAtmAnastannAmAni shRRiNuShva me. sanakaH sanAtanashchaiva prabhuryashcha sanandanaH,
vibhuH sanatkumArashcha nirmalo nirahaMkRRitiH.
There also four sons will be born of me [Shiva], all aspiring after the Yogic path and noble-souled. Please hear their names from me. They are Sanaka, Sanātana, Sanandana and Sanatkumāra freed from sin and egotism.
sanatkumAraM cha RRiShiM sarveShAmapi pUrvajam
He [Brahma] created the sage Sanatkumāra too, the eldest of all.
This leaves the Shiva purAna as an unreliable source about this issue, so there seems to be no controversy between the reliable purAnas about where the kumAras come from.
If we need to resolve a problem in purAnas, we should look to a higher authority such as itihAsa. MahabhArata, Shanti parva, 12.336.39 says:
ujjagArAravindAkSho brahmaNaH pashyatas tadA
sanatkumAro bhagavAMs tataH prAdhItavAn nRpa
Then the son of Brahma, created by a fiat of his will, Sanatkumara, studied this cult
The bhagAvatam gives us another option of combining origins: the kumAras here are depicted as some type of avatAr of viShNu (maybe just an empowered person instead of a full incarnation. I have already quoted 1.3.6 as showing that they are sons of brahmA. It says:
devaH kaumAraM AshritaH chachAra dushcaraM brahmA brahmacaryam
The lord manifested as Kumaras under Brahma, as (celibate) brahmacharis which is very difficult to do.
Although the purAna affirms that they were born of brahmA, they were also manifestations of the lord, just as rAma was born of kauShalya, but was completely the lord. This is different from the Shiva purAna 3.4.21, which actually says that they were born of Shiva, "putrAshchatvAro" is the verb for being born of someone. The connection between brahmA, the lord and the kumAras is also seen in 2.7.5, where brahmA has to perform penance for him to appear as them. This does not contradict them being born from brahmA, but the Shiva purAna statement does. BhAgavata uses the verb "’bhUt" instead in 2.7.5. RUpa GoswAmI in the laghu-bhAgavatAmRRita (1.4.36) quotes a statement from the padma purAna that I have not been able to find:
AviShTo 'bhUt kumAreShu harir vibhuH
The almighty Hari imparted his powers to the Kumaras.
In short, the kumAras were born of brahmA, not incarnations of Shiva, and possibly also incarnations of viShNu or kRRishNa.