Hello plaur
You appear to be at least a very good beginner. Just because you 'know' that
the decision is close here, you show good judgement.
Your methods are fine for your level. You should always chose methods that work for you. Playing 'better' methods badly is often the short road to trouble.
That being said, I use different methods than your current methods. I learned my method from a 'better' playar many years ago. Direct 3H raises would be 'anything' extra added than the example hand. Bidding 3H after passing is 'clear cut' in my opinion, 'if' I am playing with a good partner.
Bidding opposite a first time partner, who was a beginner, I would pass the second time. Beginners seem to 'forget' that you first passed and 'only' later raised. If I was playing opposite a player of your ability, I would pass and later bid 3H.
I hope that I am not confusing you here. I rate your game(from your questions and comments, as being beginner++ or even at the intermediate level) so I would suggest that you bid a delayed 3Hs here playing 'with' a partner that is equal to your current level of play.
TDs are not perfect. You may appeal their decision unless it is a small club game 'without' a committee available. In large club games or tournaments, you should have appealed this decision. The first pass was 'very close' to a 3H call.
The delayed 3H bid is simply good bridge in my opinion.
A committee would very likely rule in your favor here(committees are also not perfect)
Regards,
Robert