No, ammonia is not the main product of chlorination of algae. I think what you read is that you may measure some Combined Chlorine (CC) when algae is present and gets chlorinated, but even this does not always happen. Chlorine will combine with some organics (some amino acids including some in proteins including chlorophyll and some in DNA) and oxidize some of them which is how algae gets killed. Some of these may measure as CC until further oxidized, but usually such products stay in algal clumps that need to be filtered out.
None of the common pool chemicals have ammonia in them. The only exception is one type of algaecide (ammonium chloride) that intentionally adds ammonia in order to combine with chlorine to create monochloramine that kills algae and does not get moderated in its strength by CYA so can work around high CYA levels.
There are two types of tests for ammonia. The saliclyate test works by adding some chlorine (usually from Dichlor) to form monochloramine and then measures monochloramine. So this test is obviously interfered with by monochloramine. The test that uses Nessler's Reagent reacts directly with ammonium ion (though being in equilibrium with ammonia it measures total ammonia). Remember that if there is ANY chlorine present, then any ammonia will very quickly be converted to monochloramine. So the only time an ammonia test kit is useful is when there is 0 ppm FC such as on spring opening. In any event, testing for ammonia does not tell you that algae is present.
The best way to know if algae is present is to either 1) see it or 2) perform the
Overnight Chlorine Loss Test (OCLT). A loss of chlorine overnight could just be oxidizing organics in the pool, such as from blown in leaves or pollen, but if one doesn't see any of that then its likely to be nascent not-yet-visible algae.
Green algae is fairly easily killed by chlorine. It is yellow/mustard algae and black algae that are more resistant though can still be killed but using higher chlorine levels and using special techniques such as getting behind light niches for yellow/mustard algae and using a heavy brush to scrape off the waxy surface layer of black algae.
Why are you asking about chlorine-resistant algae? In
this post it sounds like your chlorine level is high. You said you had a green tint to the water, but you were using copper ionization and copper that is oxidized and when the pH is higher is green. There is no mystery as to what is going on in your pool. You need to rip out (disable) the copper ionization, test your pool water for copper levels, and drain/refill to lower the copper levels substantially if they are high. It doesn't sound like you have any sort of algae problem. Quite frankly, since it's only a 2800 gallon Intex pool, I'd just start over with a fresh water fill and again, stop listening to pool stores and stop using metal ionizers.