As no one else has substantively answered, a partial answer (to long for a comment) -
Section 17(1)(g) act says that No person Shall "spit any substance or expel mucus from the nose upon or onto any street or any public place...". Somewhat unhelpfully, a public place is defined "includes any place whether privately owned or not to which the public has access;"
The rest of this post is opinion only -
From a purely practical POV, I can't see spitting into a toilet as being a major issue - just make sure no one actually sees you just to be safe and flush afterwards - Singapore have fines for not flushing toilets as well !)
You may or may not realise it, but when I was last in China many years ago, people routinely simply spat on the streets due, in part, to the pollution - and I surmise this is the activity the law seeks to curtail. I have observed that since coming to a country without that level of pollution, this habit generally goes away - I know a person who used to do this who no longer did a few months after immigrating. I believe it is this behaviour - done in public - that Singapore is trying to eliminate - not that it was that polluted.
The same Section 17(1) says that "no person on shall deposit, drop, place or throw any dust, dirt, paper, ash, carcase, refuse, box, barrel, bale or any other article or thing in a public place" - which makes me wonder if a bin placed by the council would actually be a public place !