"Debt issuer default risk" and "counterparty risk" are very similar. From Risk magazine:
Counterparty Risk
The risk that a counterparty to a transaction or contract will default (fail to perform) on its obligation under the contract. Counterparty risk is not limited to credit risk (the risk that the counterparty cannot fulfill its contractual obligations for payment) but may also result from other problems associated with a counterparty unwilling to honor the contract.
In practice, it is implicitly assumed the material part of CVA can be captured by CDS prices. Quantifying other sources of counterparty risk would be particularly challenging due to the absence of legal precedents (bear in mind that the CDS market is only 25 years old).
Now suppose you have entered into some interest rate derivative contract with your counterparty $A$. The contract has maturity $T$ and its value at $t<T$ is given by $V(t)$. Your exposure $E(t)$ at time $t$ is equal to the positive value of the contract:
$$E(t)=\max(0,V(t))$$
CVA is the price of the risk of default of your counterparty thus you need to look at the future expected exposure under the risk-neutral measure. Enter the risk-neutral discounted expected exposure $EE^Q(t,u)$ for a time $t<u<T$:
$$EE^Q(t,u) = E^Q_t[D(t,u)E(u)]$$
where $Q$ is the risk-neutral measure and $D(t,u)$ the discount factor from $u$ to $t$. This corresponds to the risk-neutral value at time $t$ of the future value to you of the derivative contract, namely what you stand to lose if your counterparty defaults (or is "unwilling to honor the contract"). The default can occur at any time $u$ between $t$ and $T$, hence CVA for counterparty $A$ at time $t$ is equal to (assuming a recovery rate $\text{Rec}$):
$$\text{CVA}_A(t)=(1-\text{Rec})\int_t^TEE^Q(t,u)\color{blue}{\text{d}P_t^Q(u)}$$
where $\text{d}P_t^Q(u)$ is the risk-neutral probability of default of $A$ on the infinitesimal time interval $[u,u+\text{d}u]$, conditional on the current ($t$) information.
To obtain estimates of default probabilities, you can extract information from the CDS market for counterparty $A$. To understand this, consider the following toy example: assume the risk-free rate is constant equal to $r$ and there exists a contract with a $1$-year maturity which pays $1\times(1-\text{Rec})$ if counterparty $A$ defaults within the year in exchange for a CDS premium $s_A$. By risk-neutral theory:
$$\begin{align}
s_A &= E^Q[e^{-r}(1-\text{Rec})1_{\{A \text{ defaults within 1 year\}}}]
\\[3pt]
& = e^{-r}(1-\text{Rec})\color{blue}{P^Q(A \text{ defaults within 1 year})}
\end{align}$$
Thus:
$$\color{blue}{P^Q(A \text{ defaults within 1 year})}=e^r\frac{s_A}{1-\text{Rec}}$$
You observe that CDS premiums encapsulate expected default information. You could also consider that distributional information $P^Q(\cdot)$ contained by CDS premiums is not limited to default but includes any failure to pay: this would probably depend on the design of the CDS contract and its payment triggers.
Anyway, in practice to price CVA we only look at the probability information contained by CDSs and assume it captures any material event that might result in no payment.
[Edit] In practice you also need to consider which CDS you choose: different CDSs might have different contractual designs and definitions of "default", thus some CDSs might not be relevant to quantify the probability of default/no payment/etc. for a particular derivative contract $V(t)$. See comments from @Mehness.
References
Pykhtin, M. and Zhu, S. (2007). "A Guide to Modelling Counterparty Credit Risk", GARP Risk Review.
