The other issue with translating adam is the use of the definite article h'adm 'the man'. adm without the article can mean a man, people in general, mankind or it can be the personal name Adam. Most of the time on the beginning of Genesis, certainly most of Gen 2 and 3 you have h'adm, the man, not the personal name Adam.
The first three references are adm, but are translated man because they seem to refer to mankind in general. Gen 1:27&28 "Let us make man in our image" but it is talking about male and female rather than an individual male. Gen 2:5 is also translated man, because it seems to be referring to man in general, there was no man to work the ground.
After that all the references in chapter 2 have the definite article, except 20b But for Adam there was not found a helper fit for him. Again this could also be translated and to man hath not been found an helper--as his counterpart (Young's). Genesis 3 again uses 'the man' except for 17 And to Adam he said, "Because you have listened and 21 And the LORD God made for Adam and for his wife garments of skins. Again Young is quite happy to translate these the man.
The AV tended to translate a lot of the h'adm as Adam where more modern translations, especially the more literal ones, say 'the man'. The AV (and NKJV) have Adam 11 times in the first 3 chapters starting in Gen 2:19.
The NIV has Adam in 2:20b and 3:17,20&21.
The NLT uses it in 3:17,20&21.
The ESV and NASB uses it in 2:20b and 3:17&21.
The RSV has Adam in Gen 3:17&21.
The tendency it to translate the three references without the article Gen 2:20b and 3:17&21, as Adam, though not always 2:20b. Meanwhile less literal translations also throw in Gen 3:20 probably because it is referring to the man naming his wife Eve.
The first time the NRSV, Young's Literal Translation or Greens Literal Version use Adam is Gen 4:25 And Adam knew his wife again.
Again, as glaudys says this is simply discussing grammar and translation issues, not theology.