The Scot was asked about the position after Crystal Palace announced the appointment of Iain Moody, who was Cardiff's former head of recruitment before being suspended by the club on October 9, as their new sporting director.
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"I'm the Cardiff City manager and I said that yesterday," he said at his press conference. "I’ve got a great group of players and staff. I’m the manager of Cardiff City and I’m proud to be that.
"I thought our back five - our two centre-back especially - were immense today, they were playing against a real handful in the two players they were up against. But I thought they were both immense.
"He's [Caulker] a 21-year-old and he’s captaining a team in the Premier League. If he keeps playing like that, there will be plenty of England caps coming for him."
Paul Lambert, meanwhile, saw Leandro Bacuna's free-kick from distance open the scoring to give his side their first goal in over seven and a half hours of football and says he was always confident his side could score.
"I was never too down about it because I knew, against Everton, we created chances and I knew, against West Ham, we created chances," he admitted. "I knew we would create, it’s just [to] keep persevering and keep doing what we’re doing.
"I thought they were excellent. When you look at the side we put out, we have a lot of injuries at the minute – we had half a team out. I thought we were worthy winners.
"I thought his [Bacuna's] free kick against Manchester City was world class but I thought that one surpassed it. It was a terrific free kick. But Leo, as I’ve said all the time, can do it."