jfpe,
If I understand this correctly, you saw a former coworker's trumped up resume on a networking site where people with his qualifications advertise for work.
1. Your complaint is that he said he ‘managed’ a project where you think he did something less, maybe ‘assisted with’ or ‘participated in’ the project.
2. You want to do something, perhaps make a formal report, to let others know that it is your opinion that he did not manage the project and that your former coworker is less than competent.
3. Your motivation for this is that it may possibly impact your ability to get a project for which you are both competing.
4. Or, he may have gotten a project on which you submitted a bid, or fee estimate, and now you want to get him off the project.
If you are in the US, you may be opening yourself up to a legal claim by your former junior coworker for tortious interference, especially in the case where he won a contract. You, a third party, is attempting to interfere with someone’s business or contractual relationship with another even though you have no legal claim to the contract for the project.
And so what if he said ‘managed.’ You can manage a job well, or you can manage it badly. In the end you will be arguing semantics. Maybe this former coworker considered what he did to be a managerial role and the people who you considered the true managers, he is calling supervisors, or overseers. He wants to be called the ‘manager’ for the project and you want him to be called ‘assistant incompetent junior almost engineer’.
My suggestion, drop it.
If competence is important to the people doing the awarding, they will look for someone else. If they hire your coworker because he is cheaper and don't care about the competence, it's their decision, not yours.
"If you are going to walk on thin ice, you might as well dance!"