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Cook, 32, resigned in February and in his first match under replacement Joe Root made 69 as England defeated South Africa by 211 runs at Lord's.
"I was having a laugh watching him in the field, running around in the covers," said Bayliss.
"He looked like he was enjoying it. For someone at this stage of his career, that's a good sign."
Like Cook before him, Root began his reign as captain with a victory, scoring 190 in the first innings to help rescue England from 76-4.
At 26, the Yorkshire batsman had only captained in four previous first-class matches before taking on the England job.
"Before this Test he was a little more on edge than I've seen him before," said Bayliss. "We spoke about Joe being himself, getting the message across in his way.
"He's proactive and confident and I thought that came across in the way he led the team.
"Root and Cook are different characters, they go about their batting in different fashion, so the dressing room was a little different. A lot of the messages were similar, just delivered in a different way."
England have named an unchanged 12-man squad for the second Test of the four-match series against the Proteas at Trent Bridge, which begins on Friday.
That means the balance of the team is likely to remain the same, with left-arm spinner Liam Dawson continuing alongside Moeen Ali, who took 10 wickets at Lord's.
"The combination we had you could pretty much pick on any type of wicket," added Australian Bayliss, who further clarified Moeen's role in the team after saying on Sunday he thought his man-of-the-match performance could be a "watershed" moment.
"He wants to be in the team as a batter who bowls a little bit, so that is what we have chosen him as," said Bayliss.
"Mo is a bit of a complex character at times, so it's more for his benefit than anything. It takes the pressure off him. It doesn't mean he can't be the best spinner, but his number one job is to bat.
"This Test was probably a sign that this is the best way to go with him."
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By BBC cricket correspondent Jonathan Agnew
Alastair Cook looks like has he gulped in a huge breath of fresh air by giving up the captaincy.
I interviewed him on the fourth morning and was able to draw a comparison between the Cook of now and the Cook I spoke to after the fifth Test against India in Chennai last year. It was like a different man.
Cook will not be a grumpy old pro, chuntering in the corner as the team move on. He will enjoy his cricket and bat the way he does. The 69 he made in the second innings was classic Cook - playing nicely without taking the attack apart.
With his schedule - no one-day internationals - he could play Tests for a long time to come.
Read more from Jonathan Agnew here.

Mary Boyle was six years old when she went missing on 18 March 1977 whilst visiting her grandparents at Cashelard.
The dig was prompted by Danish psychic Yan Sterns, hired by country singer Margo O'Donnell - Daniel's sister - a close friend of Mary's mother, Ann.
Gardai have confirmed they were assisting the family in the fresh search near Ballyshannon.
Material taken from the ground is to be examined.
Mary Boyle is officially the Republic of Ireland's longest missing person.
The psychic visited a remote hillside near Ballyshannon on Friday and pointed to where the dig should take place.
It was his third visit to the area.
Gardai supervised the dig. They said the would act on any information supplied.
Mary Boyle disappeared 34 years ago but this is a still "a live investigation", a spokesman for the Irish police said.

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2 November 2014 Last updated at 10:11 GMT
The BBC's Ireland correspondent, Chris Buckler, reports .
Opener Alastair Cook is "loving" life without the England Test captaincy, says head coach Trevor Bayliss.
[NEXT_CONCEPT]
A fresh dig to find the remains of a young girl who disappeared almost 34 years ago has taken place in Donegal.
[NEXT_CONCEPT]
As the UK considers greater devolution in the aftermath of Scotland's independence referendum, should a troubled Northern Ireland Assembly push for more powers over its own affairs?