Note that the message about reinforcements means at least one 'wave' has spawned in. It's not a general 'Reinforcements are now activated!' alert. You can get this several times.
I've even done tests by reloading saved turns and not entering the manor (or going into the back yard of the manor).
If this is just a coincidence than I am extremely surprised.
Reloading means relatively little, since any given turn has at least ~2/3 chance of not getting reinforcements. So if you reload, chances are reinforcements won't reappear. I just reloaded six times in a row and they didn't respawn.
I think what's happening is that your mansion door assaults coincide with turn 3-5. I don't think triggering reinforcements by entering a specific map block is even possible. You could spawn enemies with something like land mines, perhaps, but they'd just appear next to where the mine was.
Try a manor assault, breach the front door, save and reload several times. Or try a tier 2 manor, blow the roof before turn 5 and enter. I think you'll see that there's no correlation.
Edit: Or you could just edit alienDeployments_XCOMFILES.rul, find
reinforcements: &EliteSoldiersAndServantsGreatestManorReinforcements
- type: servants
...
- type: northwave
executionOdds: 10
minTurn: 3
maxRuns: 1
...
- type: southwave
executionOdds: 10
minTurn: 3
maxRuns: 1
...
- type: eastwave
executionOdds: 10
minTurn: 3
maxRuns: 1
...
- type: westwave
executionOdds: 10
minTurn: 3
maxRuns: 1
...
and replace all of these with
- type: [whatever]wave
turns: [0]
maxRuns: 1
...
You'll discover that all the reinforcements have already taken up positions regardless of what you do to the front door.
This is also true for "Golden Towers".
Golden Towers have pretty much the same reinforcements as manors.
Edit2: I think this is another case of something random with an explanation someone finds reasonable and will just not believe it doesn't work that way. Some people have been swearing up and down that they could stop infiltrations by shooting down the Battleships, or that downing a UFO with a Cannon instead of missiles increases the chance of leaving the power source(s) intact.