Is it known if Yeltsin was planning to give up power peacefully if he lost the 1996 election?

As I understand from another (2001) source, the 1995 WaPo story quoted in the Q is somewhat strange. First, it's worth noting that by "hard-liners" the WaPo story means war hawks who were pressing for a continued offensive in Chechnya.

Besides diverting some IMF funds, Yeltsin won because he made a deal with 3rd-placed candidate Lebed, before the run-off. At that time, Lebed favored a ceasefire in Chechnya to give the Russian military time to reorganize. And Yeltsin went along with it. It was apparently defense minister Grachev who disagreed and might have been inclined to stage a coup, but he was replaced with a Lebed supporter. (Grachev was largely responsible for the badly organized attacks from the previous years, having e.g. dismissed theater commanders who were asking for more time to prepare.)

The WaPo story, which was written in 1995, before the events from the above para took place, actually did venture a guess that that was an alternative route (to a coup):

Short of declaring an emergency, there are signs that Yeltsin may try to deflect blame for the war by finding a scapegoat in his government or among his top advisers. The most likely prospect is Defense Minister Pavel Grachev, a deeply unpopular figure whose performance in recent weeks has widely been considered incompetent.

So yeah, instead of staging the "silent coup" with support of the Chechnya war hard-liners, which some "some analysts" thought was going to happen, Yeltsin went for dumping (one of) them war hawks hardliners, as politically convenient at the time.

Now it's also true that some years prior, in 1993, Yeltsin dissolved parliament (well, the Supreme Soviet), but the dissolved parliament met anyway and impeached and dismissed him. Yeltsin then declared a state of emergency and surrounded the parliament building with troops, but also organized early parliamentary elections. So, the analysts who were thinking he was going to declare a state of emergency again in 1996 were perhaps basing their guess on an analogy with 1993... although in that prior case his actions were supported by Western leaders. It's harder to see how doing the same after straight losing an election would have kept that kind of external support (and money flowing) even though the West was quite worried what the communist Zyuganov might do if elected.

If you want an alternative account, Korzhakov, who had a fallout with Yeltsin, later wrote that:

In regards to the presidential election in 1996, as well as in 1993, Yeltsin had – according to Korzhakov – no intention of leaving the Kremlin and was prepared to take any actions to insure his continued stay. Yeltsin felt free to express his contempt toward democratic principles in Korzhakov's presence. Some episodes the author relates are almost resonant of the Nixon years and the published Watergate tapes. According to Korzhakov, in the summer of 1996, Yeltsin and his prime minister, Victor Chernomyrdin debated the very serious possibility of canceling the presidential election. Korzhakov adds several significant details which possibly confirm the view that the Kremlin violated many democratic rules during the presidential election campaign in 1996.

I'm not sure how much of this has been corroborated. The same Wikipedia article notes that "On 5 May 1996, Korzhakov explicitly called for postponing the elections."

In a 1998 review of Korzhakov's book (which is described in rather unflattering terms as "highly partisan" and "a study in sour grapes") the following rather different take is presented:

Ideally, as far as Korzhakov was concerned, there would have been no elections at all. Failing that, he had another strategy about which he is uncharacteristically coy. By ideological persuasion, he, Soskovets and their friends were ‘gosudarstvenniki’ – believers in a strong protectionist state. That brought them close to the nationalist wing of the Communist Party. Korzhakov confirms that he had meetings with one of the Communist leaders, Viktor Zorkaltsev, to whom he proposed bringing the Communists into a ‘coalition government’. The skeleton of an idea was beginning to emerge: maybe a deal could be struck which would allow Yeltsin to carry on as President, without an election but with much reduced powers; the ‘democrats’ would be shut out, the Communists would be let into government and the likes of Korzhakov and Soskovets would be the kingmakers.

We don’t know how far he got with this mad scheme, but it was treasonable to Yeltsin and his people. Korzhakov had not really paid enough attention to the election campaign proper, which Yeltsin was now beginning to win. The election went ahead, and on 20 June, between the two rounds of voting, after a night of intense feuding between the two camps, Korzhakov, Barsukov and Soskovets were all sacked.

There may a bit more to the story. While the two factions in Yeltsin's camp were still fighting each other, the communists, who had majority in the Duma, staged a sort of legal coup, rolling back the agreements that ended the Soviet Union. This made Yeltsin consider suspending their party for two years, before the elections... according to Korzhakov, anyway. But after some heated talk with Chubais and apparently also on the advice/opposition of his daughter, Yeltsin changed his mind after a couple of days and didn't go through with the plan to suspend the communist party before the elections.