I'm right now considering joining the Imperial Legion, but one thing I would like to make sure is that I'm probably seen as being part of it while I'm (at least) wearing the armor... So do other Imperial Soldiers adress you like being part of the legion?

yup

In my game many legioners call me as Legate (example), but legioners who follow a prisoner not identificate me as soldier of Legion. They call me "citizen" or some... I don't know right words, because play in localised (non English) version Skyrim.

P.S. I'm not wearing Imperial armor.

If you wear the imperial armor and havent joined the imperial cause and goto stormcloak encapments and talk to the soldiers they will comment on your armor and you get dialogue options to say you support the empire. They will then turn hostile and you have to deal with the "essential" commanders that cannot die. That was fun to find =)

Once I joined the legion, only done a couple quests, I can still walk amongst the stormcloak camps, even go stand by ulfric himself and nothing happens. havent tried talking to him tho lol.

Unless you wear the armor?

I did this with the Stormcloak camp up in the Pale. Strolled right in fully decked out in heavy legion armour, declared I was with the Empire and then was chased out by about 6-7 Stormcloak soldiers. I led them all to the Empire's camp slightly north and was pleasantly surprised as about 5-6 Imperial lads came running out to meet them. Was one of the biggest fights I've been in. Good times.

Good thing you were there to help them. In all of my random encounters between 'Cloaks and Legion, the 'Cloaks always win. It's those 2-handers, I tell you.

Yes, it's the two-handers - and since the Legion soldiers don't have shields to compliment their one-handers, they don't stand a chance.
Luckily, there's a mod that gives them shields to even the odds out a bit.