What is it called when a person pronounces the letter t in the word "metal" as something more similar to a d sound? And what is it called when a person stresses the t in the word "metal" to be more clearly like a regular t than like a d (like the difference between how an American says "metal" and how an Englishman might say that word)?
The phenomenon is known as "flapping", and the result, transcribed as [ɾ], is a "flap". It also applies to /d/, but people notice it most when applied to /t/ since the result is more different compared to /d/. You might call is a "fast d".
If an American were to say [mɛtʰəl] very carefully, that could be called hyperarticulation, that is, aiming to stop a rule of the language from applying by slowing your speech down. I have learned to suppress flapping when I am eliciting linguistic data from people who don't speak American English, they get confused if I pronounce "metal" or "medal" ordinarily. It is generally difficult for Americans to suppress flapping. If you're asking what it's called when British English speaker don't flap, that's just "talking", there's no term for not applying a rule that is not in your dialect.
Flapping applies to intervocalic syllable offset alveolar stops. This formulation assumes some things that are less than obvious. For one thing, it assumes that syllabic and glide r count as vowels, to account for flapping in such words as "dirty" "barter" ?"cortical". And that the off-glides of diphthongs also count as vowels: "routing" "boating" "goiter".
Another assumption: syllabic resonants r, l, m, n are derived from phonemic forms with preceding vowels, which have been lost: "border", "little", "bottle", "bottom", "button" (this last only in dialects where the t does not become glottal stop).
It's also assumed that intervocalic consonants before unstressed vowels belong to the syllable of the preceding vowel. The t of your example "metallic" is in the wrong syllable to be flapped, since t is in the syllable of the following stressed vowel.
The phenomenon you're talking about is called T-flapping (and its opposite is, I suppose, the absence of T-flapping).
Many Americans pronounce the
/d/ sounds ("alveolar stops") as something linguists write as
[ɾ] (an "alveolar flap") when they come between two vowels. Most Brits don't do this.