Edited my post to clarify with some evidence.
Trump will be disqualified by a simple majority in senate. Therefore, he won't be eligible to run for 2020. Therefore, he won't win a second term.
Even if you're convicted by the Senate you're still able to run for a third term -- as he wouldn't be term limited at that point, just removed from office. Not saying he would win or anything, but just pointing something out.
This is only possible if the Senate doesn't hold a disqualification vote -- meaning that the Senate doesn't vote on if he's able to run for President again.
This happened in the past to Alcee Hastings, a former federal judge from Florida who was impeached and convicted from his post by the Senate by accepting a $150,000 bribe in return for a lenient sentence for a particular individual. Due to the fact that the Senate hadn't held a disqualification vote from another Federal office, he had been able to run for the House and had won his election -- he's been serving since 1993 in the House.
He's also the #1 Ranked in the House for Nepotism claims, see below quote from Wikipedia
In 2012, Hastings was ranked #1 out of the 435 members of the U.S. House of Representatives for paying salaries and fees to family members, according to the conservative watchdog group Judicial Watch.[30] A state-by-state report on members of Congress published by the Citizens for Responsibility and Ethics in Washington reported that Hastings paid his girlfriend, Patricia Williams, an attorney who worked as his deputy district director, $622,574 over the four-year period, from 2007 to 2010.[31]
So yeah, fun little tidbit of knowledge.
As I understand it a simple majority in a Senate impeachment trial means nothing. 2/3rds will vote to convict on one or more articles and the president will be removed from office, or the president will be acquitted. These are the only two possible outcomes.
If however, the president is convicted, I think it's up for debate on whether or not he's eligible to run again.
From the constitution:
Judgment in Cases of Impeachment shall not extend further than to removal from Office, and disqualification to hold and enjoy any Office of honor, Trust or Profit under the United States; but the Party convicted shall nevertheless be liable and subject to Indictment, Trial, Judgment and Punishment, according to Law.
So if he's convicted, I don't think he could even run for the Mayor or West Palm Beach if he wanted. But he can't be disqualified by a simple majority.