Is there a definition of the pseudo-random sequences like PRG or PRF in cryptography? I have learned the definition of PRGs and PRFs. As I know the meaning of pseudoreandomness. I think that the pseudo-random sequence and the truly random sequence are indistinguishable.

I did not find a definition of the pseudo-random sequences. And I think the definition should describe some possible collections of "pseudo-random sequences" (e.g. M-sequence).
Follows are my approaches which may be helpful.

Let a sequence $\alpha = (a_{1}, a_{2}, \ldots, a_{n}, \ldots)$ is a sequence over a finite set $\Sigma$ ($|\Sigma| &gt; 1$), where $a_{i} \in \Sigma$. It is truly random if $a_{i}$ is chosen uniformly in $\Sigma$.

Let $\beta= (b_{1}, b_{2}, \ldots, b_{n}, \ldots)$ be a sequence over $\Sigma$. It is pseudo-random sequence if for every PPT algorithm $A$, there is a negligible function $\varepsilon$ such that

$$\left\vert \Pr \left[ A(b_{1} \Vert b_{2} \Vert \cdots \Vert b_{n}) = 1 \right] - \Pr\left[ A(a_{1} \Vert a_{2} \Vert \cdots \Vert a_{n}) = 1 \right] \right\vert \leq \varepsilon(n)$$

Or can we consider that it is unpredictable? Let $\beta= (b_{1}, b_{2}, \ldots, b_{n}, \ldots)$ be a sequence over $\Sigma$. It is pseudo-random sequence if for every PPT algorithm $A$, there is a negligible function $\varepsilon$ such that

$$\left\vert \Pr \left[ A(b_{1} \Vert b_{2} \Vert \cdots \Vert b_{n}) = b_{n+1} \right] - \frac{1}{|\Sigma|} \right\vert \leq \varepsilon(n)$$
I think that the pseudo-random sequence and the truly random sequence are indistinguishable.
And I think the definition should describe some possible collections of &quot;pseudo-random sequences&quot; (e.g. M-sequence).

Congratulations, you have a very good intuition! You're almost there.
A key to understanding the definition of a pseudorandom sequence is that there is no such thing as a pseudorandom sequence. If you have a specific sequence, then there's no randomness involved! Instead, the useful concept is that of a pseudorandom sequence family, more usually called pseudorandom function family (but abbreviated PRF). In a nutshell, a pseudorandom sequence family is a family of sequences such that given some data that may or may not be part of a sequence in that family, and bounded computational power, there is no way to know whether the data is from a sequence in the family or was generated uniformly at random. In other words, a pseudorandom sequence is indistinguishable from a random sequence.
Note that “uniformly at random” here is a mathematical definition: a uniformly random bit sequence is one where the value of each bit is independent (in the sense of probability theory) from the value of other bits as well as from all other events in the world. This definition does not consider whether it is possible to perform this generation in the physical world.
The definition of pseudorandomness only considers the output of the process, not how this output is produced. Otherwise it would be a circular definition: a pseudorandom generator is one that produces pseudorandom output — “pseudorandom output” has to have some definition other than where the output come from. The definition is about what you can deduce from observing the output. A pseudorandom generator is an algorithm that transforms a seed into the element of a pseudorandom sequence family that is indexed by the seed value.
To put this indistinguishability in more mathematical terms, consider an observable property $A$ of the output. For example, “bit $3$ is 1 and bits $4$, $5$ and $6$ are not all equal”. This observable property has a certain probability $\Pr_U(A)$ of being true for the uniform random distribution ($\frac{1}{2} \cdot \frac{6}{8}$ for the example above). If you pick a random element of the pseudorandom sequence family $F$, the probability of this observable property $\Pr_F(A)$ must be close enough to the probability for the uniform random distribution: $\Pr_F(A) \approx Pr_U(A)$.
To make this definition fully precise, we still need to say what observable properties are permitted and what “close enough” means. We aren't interested in adversaries with infinite computational power, which is why we restrict $A$ to polynomial computations. And “close enough” is defined as bounded by a negligible function. That's how the equation in your question comes about. The piece that you were missing is that you don't first select specific sequences $\alpha$ and $\beta$. Rather, $\alpha$ and $\beta$ are a probability distribution over sequences: $\alpha$ is a choice of element of the function famility, and $\beta$ is a uniform choice over all possible functions.