It appears you're using the word one as a synonym of some thing exists.
Implicitly you are not asserting that it is everything - in Aristotles terminology, the All; whose concept includes existence.
Hence it must be individualisable - in the sense of being distinguishable from everything else; and this too is part of your concept of one.
In this bare sense of one, you are not asserting that it is unique - there may be others - in which case it would be 'one of'; but it also might be the case, that it is in fact unique - ie 'the only one'.
Mathematically, one is not bare in this sense; it has a range of meanings:
Operationally as the identity
Order-theoretic: it is the first
Change: the least difference
These properties define the Peano axioms; which is the formalised concept of the positive integers; arguably it is this context that we mostly think of when we think of the number one.
None of these properties are contained in the concept of the bare one. Hence, they are definitely not identical.
The standard symbol for 'there exists' is a reversed E; it may it may not exist uniquely - if it in fact does exists uniquely, then sometimes an exclamation mark follows the reversed E.