No, it doesn't say so.
Words mean things, and that is especially true for laws and contracts.
"No person shall be elected to the office of the President more than twice..."
If they would have just stopped there, added "for any reason whatsoever", this would all be over. But noooooooooo. Therefore :
"and no person who has held the office of President, or acted as President, for more than two years of a term..."
This means the amendment applies only to those who have been President for more than two years. Left unsaid - are those two years cumulative, or consecutive? We do not know because the SCOTUS has never ruled on the meaning. A very good legal case can be made it was intended only to apply to those who had served two consecutive years as President. If you remove Trump from the Presidency the clock stops. If Pence appoints Trump as VP then resigns, the clock starts running again. In other words, it is just as I have represented it - the 22nd Amendment does not apply as Donald Trump has now been President for less than two years.
While we are wandering around this neck of the woods, you should see the inherent YUGE flaw by now - that being the 22nd Amendment does NOT apply to those appointed as President. Show me the wording that prevents someone from being *appointed* President again and again and again. The word is "elected", not "appointed". That's why the 22nd was, is and always will be the most deeply flawed amendment.
To continue :
" to which some other person was elected President shall be elected to the office of President more than once."
Again, this hinges on someone else being elected, and then in turn YOU being **ELECTED** President. Nothing here speaks to being ***APPOINTED*** President.
You just can't read these things for what you want them to say - they have to be read as they are, just like all contracts. At best, the 22nd Amendment controls normal terms of office, and run-of-the-mill elections. The second one steps out of those constraints, all bets are off. Not because I say so, but because there are loopholes one could drive a semi-tractor through with no problem, because of what the 22nd Amendment DOESN'T say.