How to pronounce a 't' in American English

Allophones of /t/, continued.

Other symbols that will be used in a chart to follow:
  V'        stressed vowel      a vowel relatively more stressed
                                              syllable of the word
  'V      unstressed vowel   a vowel relatively less stressed
                                               syllable of the word
  V        any vowel             any vowel, whether stressed or not
   L        syllabic L              as in the final syllable of "little", "curdle"
                                            (always unstressed)
   N       syllabic N              as in the final syllable of "satin", "carton"
                                            (always unstressed)
   /        "or"
   b#     beginning of a word
   e#     end of a word
   C      consonant               any (other) consonant
                                           not mentioned in a list
                                          (down the left side or
                                           across the top of the chart)

To be continued.
CJ

Explanation of the chart with examples.

Before or after s (or f), t is neutral (first row; first column).
Other than the case above, t before r or a stressed vowel is aspirated.  (cols. 2, 3)
Word-initial t is also aspirated.  (row 2)

A t after a vowel or after an r and before an unstressed vowel or syllabic L is tapped (and voiced).

Other than the cases above, s t after a vowel or after an r is stopped. ([.t]).

And so on.

Examples:
lasts Watson artful ants facts street ostrich story extent after extra nest  raft
t'ruck at'rophy at'rocious fort'ress ent'rance act'ress t'orrent t'eabag at'omic att'end
art'erial curt'ail ult'erior int'ernal fict'itious t'omorrow t'oday t'wist
item atom butter barter mortar little battle hurtle startle
Dalton Clinton mountain winter captain raptor invective lintel fractal
Hi.tler wi.tness hear.tless par.tner guil.tless an.tler tac.tless
co.tton sa.tin car.ton cer.tain Dal.ton Clin.ton moun.tain a.tmosphere
sa.t  coa.t  no.te  ar.t  por.t  sal.t  wan.t 
rapt  sect

(Note that words like Dalton, Clinton, mountain have alternate pronunciations.
  1.  With a neutral t followed by vowel (schwa) and consonant n
  2.  With a stopped t followed by syllabic N)

Words with multiple t's are good practice.  attit'ude  t'urtle 

This set of posts does not deal with "t" pronounced "ch" or "sh" or silent in words like picture, nation, ballet.

CJ

1.  It is not a significant mistake to interchange t sounds.  You will be understood no matter how you pronounce your t's -- as long as you don't pronounce them as m's or j's, of course!
In fact, if you don't use the tapped t at all, for example, you will simply be pronouncing more in the British style.  (We associate the tapped t with American pronunciation.)

2.  I am 98.44% certain that the neutral t I describe above is essentially the unaspirated Romance Language t (Spanish, French, Italian).  Whatever differences there are, if any, must be minimal.

3.  Your rule for tapped t is just slightly off.  Tapped t (t like d or like Spanish r) is used only in four cases, all of them between a stressed and unstressed syllable.

  a)  between vowel and vowel (intervocalic t),
  b)  between r and vowel,
  c)  between vowel and syllabic l
  d)  between r and syllabic l.

a)  bitter, meter, photograph, erratic, later
b)  barter, fortify, artifice
c)  little, natal, cattle, subtle (b is silent)
d)  hurtle, myrtle, fertile

Just saying "between voiced sounds" covers too much territory.  It's true that all cases of tapped t occur between voiced sounds, but, on the other hand, some t's between some voiced sounds are not tapped t's.

li.tmus, par.tner, Sha.tner, Hi.tler

CJ

I'm not familiar with MLB.

In American English it is very common to omit the "t"  when it follows a stressed syllable ending in "n" and precedes an unstressed syllable beginning with a vowel.  "winter" and "winner" are indistinguishable.  The absence of the t sound here is not an aspirated t, as you seem to suggest in your post.

 I don't recommend imitating this pronunciation unless you live in a community where it is very common.  Use a neutral or slightly aspirated t instead.

There is an exception -- the verb "want".  "want to" and "want a" are both pronounced "wanna" if you are speaking fairly fast in an informal setting.  The past tense is often said "wannid".  "wanted to" is often "wannida".  I don't see any great objection to that in relaxed conversation.

"sentence", by the way, can have stopped t and syllabic N:  sen.t-Nce.

CJ