Buddha's omniscience Is a Buddha omniscient? If so, is there some agreement on what omniscience more specifically means?

(All traditions are of interest. But particularly Mahayana)
In MN 71, the Buddha clarifies the meaning of his omniscience to the wandering ascetic Vacchagotta:

Vaccha: ‘Venerable sir, I have heard this said: The recluse Gotama is all knowing and all seeing and acknowledges remainderless knowledge and vision in this manner, ‘while walking, standing, lying or awake, constantly and continually knowledge and vision is established, to me’.Venerable sir, those who say, these words, ‘The recluse Gotama is all knowing and all seeing and acknowledges remainderless knowledge and vision:while walking, standing, lying or awake, constantly and continually.’ Are they saying the words of good Gotama and are they not blaming the Blessed One falsely?
The Buddha: Vaccha, those who say, the recluse Gotama is all knowing and all seeing and acknowledges remainderless knowledge and vision, while walking, standing, lying or awake, constantly and continually. They, do not say my words, they blame me falsely.
Vaccha: Venerable sir, saying how would it be said, correctly not blaming the Blessed One falsely?
The Buddha: ‘The recluse Gotama knows the three knowledges. Saying it thus they would be saying the right thing, and not blaming me falsely. Vaccha, whenever I desire, I recollect the manifold previous births, such as one birth, two births, with all modes and all details, thus I recall the manifold previous births. Vaccha, when I desire, with the purified heavenly eye beyond human, see beings disappearing and appearing, in unexalted and exalted states, beautiful and ugly, in good and bad states,-- I see beings, according their actions. Vaccha, destroying desires, my mind released and released through wisdom, here and now by myself realising I abide. Vaccha, if it is said, the recluse Gotama knows the three knowledges, saying it thus, you would be saying the right thing and not blaming me falsely’.

Ven. Bodhi further elaborates by citing the Commentary in his note in &quot;The Middle Length Discourses&quot;:

MA explains that even though part of the statement is valid, the Buddha rejects the entire statement because of the portion that is invalid. The part of the statement that is valid is the assertion that the Buddha is omniscient and all-seeing; the part that is excessive is the assertion that knowledge and vision are continuously present to him. According to the Theravāda exegetical tradition the Buddha is omniscient in the sense that all knowable things are potentially accessible to him. He cannot, however, know everything simultaneously and must advert to whatever he wishes to know. At MN 90.8 the Buddha says that it is possible to know and see all, though not simultaneously, and at AN 4:24/ii.24 he claims to know all that can be seen, heard, sensed, and cognized. This is understood by the Theravāda commentators as an assertion of omniscience in the qualified sense. See too in this connection Miln 102–7.