Your understanding pretty much correct
Mode A will not give an RA
Mode C will give a TA and RA
As one of the major factors is the closure rate of the aircrat, Ive experienced a few nusience RAs inside controlled airspace. This has usually been where one is climbing and one is descending and plan to level off 1000ft apart.
Ive also had the thing screaming its guts out at me whilst flying a visual circuit at an airfield outside of controlled airspace. (Nice to know it works!)
I recon whats probably happened is that a controller has anticipated that your intended track will result in a nusience RA and, having been previously able to confirm your level from the mode C has asked you to turn it off in order to offer you both the most expedient handling.... Id be curious to know if you were given any traffic information relating to any larger aircraft (that may have been working another frequency)?
Lastly Id like to thank you for your professionalism for using in Mode C in the first place, Im aware some hold the view that its a bit 'big brother' but it really does add lots to our situational awareness.