Your fact finding of it taking place in Anaheim is correct based on announcers. I think it is safe to assume based on uniform colors that the team on the far side is the Ducks. They featured a light color top home jersey at that time.
Placing an identification on the opposing team is more difficult given video quality. Based on this schedule, it would appear that there were 20 home games that took place in the year 1999 in the 99-00 season. This also could have been in latter half of the 1998-99 season, found here. That season featured 24 home games for the Ducks, with an additional 2 postseason games versus the Detroit Redwings. So a total of 46 NHL games were played there.
Based on the jersey colors, I believe the opposing team is the Philadelphia Flyers. The light strip across the neckline appears in only 2 jerseys that have a dark body and light number, Philadelphia and Tampa Bay. Tampa Bay, however, features a large white block on the bottom portion of the jersey which would be seen clearly even with poor video quality.
They only played @ Anaheim once in the year 1999 across the two seasons, which was this game on November 3, 1999.
Aside from the jersey aspect, the "goalie matchup" comment at the end of your second video link would support this. The Anaheim goalie Dominic Roussel had posted the highest save percentage of his season in October, winning both games in which he appeared. Philadelphia's netminder John Vanbiesbrouck closed out 12th overall for the season and posted a .908 save percentage in the month of October. This would mean that the goaltender matchup would have been considered a meeting between two strong players at that point in the season, supporting this theory.
Therefore I believe this took place a the game between PHI and MDA on November 3, 1999.