en.wikipedia.org/wiki/Pro_se_legal_representation_in_the_United_States

The sentence is incorrect because a conviction does not end all rights to publicly funded counsel.

• In Douglas v. California, the Supreme Court held that when a state provides a first appeal as of right, indigent defendants are entitled to counsel on that appeal.
• In Evitts v. Lucey, the Court reaffirmed that "[t]he Due Process Clause of the Fourteenth Amendment guarantees a criminal defendant the effective assistance of counsel on his first appeal as of right."
• Federal post-conviction practice also shows the statement is overbroad: 28 U.S.C. § 2255(g) expressly provides that in proceedings under § 2255, "the court may appoint counsel" under the Criminal Justice Act.

So while there is not a general constitutional right to counsel for every collateral attack after conviction, it is false to say that once convicted a prisoner no longer has any right to a public defender or other appointed counsel.

This sentence states a categorical rule that is not true.

In federal court, a motion under 28 U.S.C. § 2255—one of the main post-conviction remedies for people in federal custody—is not treated as a separate civil case. The Advisory Committee Note reproduced with § 2255 explains that this is possible "because a motion under § 2255 is a further step in the movant's criminal case and not a separate civil action." Cornell's Wex gives the same summary: a § 2255 motion is "a continuation of the criminal case whose judgment is under attack rather than a new civil action."

Because the article states that post-conviction motions are civil motions across the board, it misstates the law. Some post-conviction proceedings are civil in character, but the blanket statement is false.