This is a complex sugya with a lot of different opinions and I am going to try and mention some of the sources which the ideas you have cited might be based upon.

1) The Gemara end Berachos 8b states that Rabbi Shimon ben Yochai said in the name of Rabbi Akiva that sometimes a person can say Shema twice in a day, once before hanaitz and once after hanaitz, and this way he fulfils his obligation of reading both the nightime and the daytime Shema. The Gemara top 9a explains that since there are some people who are still asleep before hanaitz it can therefore be considered that one is fulfilling the Mitzva of reading the nightime Shema "when you lay down" at that time.

2) The above Halacha of R. Akiva is obviously only referring to an unusual case where for some reason one was unable to say Shema until the very end of the night. However there is a major, but different, dispute amongst the Rishonim whether the ideal time for saying Shema is before sunset or after. Tosfos 9b DH l'K"S writes that the Vatikin agree to Rabbi Yehoshua that if one did not say Shema before hanaitz one may say it until the end of 3 hours. One sees from Tosfos that the ideal time for saying Shema lecatchilah is before hanaitz.

3) However Tosfos ask on this from the Mishna in Yoma 37a that Queen Heleni made a gold lamp for the Beis Hamikdash. The Gemara 37b explains that when the sun shined in the morning the gold lamp sparkled and this way everyone knew that the time for saying Shema had arrived. This suggests that Shema should be said shortly after hanaitz, not before. Tosfos answers that Heleni's lamp was good for the general public because the average person was not able to know the exact time before hanaitz that Shema should be said but someone who is expert at knowing this time, should not rely on Heleni's lamp and should say Shema before hanaitz.

4) However Rabeinu Tam (cited in Tosfos Yoma 37b DH Amar) disagrees with the above Tosfos Berachos 9b and learns from Heleni's lamp that Shema should ideally be said after hanaitz. So according to Rabeinu Tam, Shemoneh Esreh also certainly lecatchila should be davened after hanaitz.

5) Shulchan Aruch Orach Chaim 58:1 paskens according to Tosfos Berachos that Shema should ideally be said slightly before hanaitz so that it should be completed together with its blessings at hanaitz and then one should start davening immediately at hanaitz. Mishneh Berurah #7 cites Talmidei Rabeinu Yonah 4b in Rif pages DH Ad that hanaitz is defined as the time when the sun begins to shine on the top of the hills.

[See also Moadim u'Zmanim, by Rabbi Moshe Sternbuch shlita, part 4 chapter 321]

Shavua Tov

Dovid Bloom