Are these conditions equivalent to the definition of the limit of a sequence? The definition of the limit of sequence:

We call $x$ the limit of the sequence $(x_{n})$ if the following condition holds:
For each real number $\varepsilon >0$, there exists a natural number $N$ such that, for every natural number $n\geq N$, we have $|x_{n}-x|<\varepsilon$.

Now give some different conditions:

For each real number $\varepsilon >0$，there exists a natural number $N$ and a constant $x_{\varepsilon}$ such that, for every natural number $n\geq N$, we have $|x_{n}-x_{\varepsilon}|<\varepsilon$.

My question is

Are the above conditions equivalent to $\lim x_n = x$ (the definition of the limit of sequence)?

 A: Yes, they are. It is clear that if $\lim x_n=x$, then those “different conditions” hold.
Now, suppose that the sequence $(x_n)_{n\in\Bbb N}$ is not convergent. Then it is not a Cauchy sequence, which means that, for some $\varepsilon>0$, there are, for any $N\in\Bbb N$, natural numbers $m,n\geqslant N$ such that $|x_m-x_n|\geqslant\varepsilon$. But then there is no $N\in\Bbb N$ and no $x^\star\in\Bbb R$ such that $n\geqslant N\implies|x_n-x^\star|<\frac\varepsilon2$.
A: Your second set of conditions is equivalent to the definition of a Cauchy sequence, so yes, this is equivalent to the sequence being convergent.
A sequence is Cauchy if for any $\epsilon$, we can find an interval of length $\epsilon$ which the sequence is eventually contained in. Your $\epsilon_m$ is simply the center of that interval.
A: I think that the above conditions you stated stray from the notion of limit in the sense $\lim x_n = x$, and are more useful for evaluating what type of sequence you are dealing with. Like Cauchy sequences for example.
A: Nope, they aren't, because the first one names a limit value and defines a condition under which it exists whilst the second one defines a condition only and says nothing about the existence of the limit value $x$.
As a result the first one is a definition of a convergent sequence (e.g. a real sequence of $\lfloor 10^n\,\pi\rfloor/10^n$ in $\mathbb R$ for natural $n$) and the second is a definition of a Cauchy sequence (e.g. a sequence of $\lfloor 10^n\,\pi\rfloor/10^n$ in $\mathbb Q$ for natural $n$).
