Question on Brachot 26b

Many years ago I came across a question in messechet brachot page 26b.  It is a simple question and I have asked many great Torah scholars over the last 10 years and have not received a satisfactory answer.

Last time daf yomi was learning messechet brachot Mark Zuckerberg was still running facebook out of his dorm room and twitter did not exist.

This coming Monday, August 27th, daf yomi will be learning Brachot daf 26, which means hundreds of thousands of people around the world will be focusing on the very page where my question is and social media allows me to get that question out to those people.  With so many people in the world learning page 26 on the same day, I hope that this post falls into the hands of someone who is equally bothered by this question and can come up with a satisfactory answer.

Here is the question.

The Gemara says that Avraham established shacharit.  We learn that from the place in the Torah where it says:

וַיַּשְׁכֵּ֥ם אַבְרָהָ֖ם בַּבֹּ֑קֶר אֶ֨ל־הַמָּק֔וֹם אֲשֶׁר־עָ֥מַד שָׁ֖ם

"vayshkem Avraham baboker el hamakom asher AMAD sham."  Avraham woke up early to the place where he had STOOD.

The gemara then says that we know that he was davening because the verb AMAD implies tefilah - prayer.  We know this from a verse from Tehilim chapter 106:30 that says:

 וַיַּֽעֲמֹ֣ד פִּֽ֭ינְחָס וַיְפַלֵּ֑ל וַ֝תֵּעָצַ֗ר הַמַּגֵּפָֽה 

"VAYA'AMOD Pinchas vayipalel [vateiatzar hamageiphah]"   Pinchas STOOD up and executed judgement [and stopped the plague].

The word for executing judgement is VAYIPALEL which is the same root as TEFILAH which means prayer.  Granted that prayer is a form of self judgement, but according to the simple explanation of the verse, pinchas did not pray.  He acted.  To use that verse as a proof is a stretch.

But my question is as follows: given that 106:30 is a stretch as proof that the verb AMAD means prayer, why did the author of this braitah overlook a perfectly good proof found in the very same chapter just a few verses earlier?  106:23 reads as follows:

וַיֹּ֗אמֶר לְֽהַשְׁמִ֫ידָ֥ם לוּלֵ֡י מ֘שֶׁ֤ה בְחִיר֗וֹ עָמַ֣ד בַּפֶּ֣רֶץ לְפָנָ֑יו לְהָ֘שִׁ֥יב חֲ֝מָת֗וֹ מֵֽהַשְׁחִֽית

"Vayomer lihashmidam lulei Moshe bichiro AMAD baperetz lifanav lihashiv hamato meihashchit."
[Hashem] had said that He would destroy them, had not Moshe His chosen one STOOD in the breach before Him to turn away His wrath lest He destroy them.

In the Pinchas verse AMAD does not have to mean prayer.  It could easily be taken literally to mean that he stood up.  In the Moshe verse the verb AMAD is clearly a figurative expression.  Whether Moshe was sitting or standing is irrelevant.  "Standing in the breach" obviously means that he prayed.

It is hard to accept an answer that the author of the Braitah somehow overlooked this verese.  There must be some good reason why the Pinchas verse is better.  But in over 10 years nobody has been able to give me a good reason why.

If you know someone learning daf yomi please forward this to them and hopefully this time around someone can give me a good answer.  Otherwise I have to wait another 7.5 years.