At school we were taught that the sentence She said "I love you" , being converted from direct speech to reported speech form, sounds like She said she loveD me .

The verb in the second sentence stands in Past Simple instead of Present Simple: loveD . The teachers emphasised the relevance of this rule because it contradicts the logic of our native language, Russian.

Although this rule of changing tenses in reported speech seems fine to me (she said she loveD me then, but it doesn't mean that she is still loving meEmotion: smile, I still feel like using the 'incorrect' tense in some cases.

It is when the thing that was spoken out was definitely CONSTANT or obviously HASN'T CHANGED by now. For example,

At the first lecture the professor told us that Mathematics isn't easy. instead of wasn't easy [because it is as difficult now as it was then and as will always be!]
I told her that I am a vampire only yesterday. instead of I was [because I couldn't have changed within one day!]

Is it correct?
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Comments  (Page 3) 
"when the situation described in the reported clause is a permanent/habitual situation, or still exists or is relevant at the time we are reporting it, then we use a present tense (or present perfect) if we also use a present tense for the verb in the reporting clause:
Dr Weir thinks that he spends about 5 minutes on a typical appointment with a patient.
when we use a past tense in the reporting clause we can use either a present or past tense (or present perfect or past perfect) in the reported clause:
She argued that Carl is/was the best person for the job.
They noted that the rate of inflation has/had slowed down.
Choosing a present tense (or present perfect) in the reported clause emphasises that the situation being reported still exists or is still relevant when we report it."

Taken from Advanced Grammar in Use (Martin Hewings, second edition).

I believe this may help to resolve these questions.
Anyway, I believe that what way to use depends on what you want to say.