In order to solve this problem, let me discuss the first pages of Veṅkaṭanātha’s Nyāyapariśuddhi. There, Veṅkaṭanātha explains clearly his reasons for undertaking the study of Nyāya already at the outset:

The extended Nyāya is counted among the virtuous branches of knowledge |

but this has been destroyed by those various [thinkers]. Therefore it will be here purified ||

(vidyāsthāneṣu dharmyeṣu gaṇyate nyāyavistaraḥ |

sa ca viplāvitas tais tais tato ‘tra pariśodhyate ||)

As a speculation, one might think that the fact that Veṅkaṭanātha expresses his intent already at the beginning of his work might be a way to counter objections by saying that the Nyāya he will be dealing with is not the one they might want to object to.

In fact, the first thing one notices is that he had to face important objections from people presenting various quotes (which I have not been able to trace) such as the following one:

The system of Kaṇāda (i.e., Vaiśeṣika), that of Akṣapāda (i.e., Nyāya) or that of Kapila (i.e., Sāṅkhya) |

All these systems are not active in regard to the ascertainment of the self

(kāṇādam ākṣapādaṃ vā kāpilaṃ tantram eva vā |

tantrāny etāni sarvāṇi na tantrāṇy ātmanirṇaye) ||)

The following pages start discussing the technical aspects dealt with in Nyāyavistara, from ontology to dialectics and epistemology, with the verb śudh– ‘to purify’ often recurring. Thus, it appears that Veṅkaṭanātha’s system was made of concentric circles, with Mīmāṃsā being in the inner one, Nyāya in a more external one and Sāṅkhya, Buddhism and Lokāyatas even further away.