Your interpretation is correct. The Comprehensive Rules:
514. Cleanup Step
514.2. Second, the following actions happen simultaneously: all damage marked on permanents (including phased-out permanents) is removed and all "until end of turn" and "this turn" effects end. This turn-based action doesn't use the stack.
During the Cleanup Step, you lose control of the artifact. This doesn't use the stack so can't be responded to.
514.3. Normally, no player receives priority during the cleanup step, so no spells can be cast and no abilities can be activated. However, this rule is subject to the following exception:
Normally, the new controller of the artifact wouldn't be able to activate its ability. But, because there is a triggered ability waiting to be put on the stack (When you lose control of the artifact…)
514.3a At this point, the game checks to see if any state-based actions would be performed and/or any triggered abilities are waiting to be put onto the stack (including those that trigger "at the beginning of the next cleanup step"). If so, those state-based actions are performed, then those triggered abilities are put on the stack, then the active player gets priority. Players may cast spells and activate abilities. Once the stack is empty and all players pass in succession, another cleanup step begins.
So the "Tap it." is on the stack, and can be responded to. The change of control didn't use the stack and has already happened. Your opponent can use the glare when they have priority.
It's a bit odd that if this triggered ability wasn't there, your controller wouldn't be able to tap it during your turn, but that's the way it works. Note that because of the Summoning Sickness rule, your opponent wouldn't be able to use the tap effect of any artifact you gained control of unless it had haste. But the Glare cares not for Summoning Sickness.