Indeed, if you are "permanently residing outside the Russian Federation" you are exempt from military registration ("воинский учёт"), hence, a fortiori, from military service. Here is the link to the law: https://ru.wikisource.org/wiki/%D0%A4%D0%B5%D0%B4%D0%B5%D1%80%D0%B0%D0%BB%D1%8C%D0%BD%D1%8B%D0%B9_%D0%B7%D0%B0%D0%BA%D0%BE%D0%BD_%D0%BE%D1%82_28.03.1998_%E2%84%96_53-%D0%A4%D0%97/%D0%A0%D0%B0%D0%B7%D0%B4%D0%B5%D0%BB_II#%D0%A1%D1%82%D0%B0%D1%82%D1%8C%D1%8F_8 (see last line of the header of paragraph 1, just before subparagraph 1.1)
Concretely, to benefit from this, you need to cancel your registration ("регистрация по месту жительства") at your Russian address (if you still have one), and then to register at the consulate ("консульский учёт"). (Maybe registering at the consulate automatically cancels your registration in Russia, but if I remember correctly, it does not: you have to first de-register from Russia, then register abroad, in that order.)
If I understand correctly, in order for them to send you an invitation ("повестка"), you need to register at the military commissariat first. If you have not done this, then they are not even aware that you exist. They will certainly not pick you up at the airport. Your only risk is to be caught in a conscription roundup conducted by the police, e.g. in the metro. I have heard about them happening, but have never seen one (and I have spent about a year total in Moscow). And in any case, I do not think they ever happen outside the draft period (given in @undercat 's answer).
And if you are ever caught in such a roundup, be aware that the law grants you much more protection than the police would let you think. Do your research, and refuse to cooperate to any illegal requests of the police. (For example: they are not allowed to take you directly to the military commissariat, only to the police station.)
Source: I used to be in a similar situation with a similar issue, so I did my research at that time.