There's something that always concerns me in such translations, which is the way the word 'exists' or 'truly exists' is used. 'Exist' means 'to be apart' - to be separate, to be 'this' as distinct from 'that'. Practically speaking 'what exists' refers to what in Chinese Buddhism is called 'the ten thousand things', and what can be generally designated 'the phenomenal realm'. What can be distinguished from 'what exists' is 'what is real
'. And what is real in spiritual texts, is not 'something that exists', as it is not amongst phenomena, it is not 'this thing' as distinct from 'that thing'. It is transcendent, and 'existence' is what it is transcendent in respect of. (Although in Buddhism it is also never other to the the phenomenal domain, either, as form and emptiness are always conjoined, but the point remains.)
So in many of these quotations, 'truly exists' might be written as 'real' or 'actual', i.e. it might be expressed as 'perfectly real' or 'perfectly realised' - not as 'perfectly existent'. So 'perfectly real nature', not 'perfectly existent nature'. Nothing is 'perfectly existent' as 'what exists' is always a combination of existence and non-existence and always arises on the basis of dependent origination.
The issue might not be apparent in Sanskrit or Tibetan (and I know a little Sanskrit, but no Tibetan) as it's specific to current English, where there is equivocation between 'exists', 'reality' and 'being', all of which have overlapping meanings, but which aren't necessarily synonyms. So I'm not for one minute implying any criticism of the original texts or ideas, but the way they have been translated into current idiomatic English.
A note from western philosophical theology:
"Existence" refers to what is finite and fallen and cut of from its true being. Within the finite realm, issues of conflict between, for example, autonomy (Greek: 'autos' - self, 'nomos' - law) and heteronomy (Greek: 'heteros' - other, 'nomos' - law) abound (there are also conflicts between the formal/emotional and static/dynamic). Resolution of these conflicts lies in the essential realm (the Ground of Meaning/the Ground of Being) which humans are cut off from yet also dependent upon. Therefore existence is estrangement."
Take this as a footnote.