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I have never seen any experienced climber tie up a bowline knot starting up with an underhand loop (*). Bowline knots are most typically started with an overhand by flipping the standing part of the rope under the running end.

After the comment by Ben Crowell, it is worth pointing out that starting with an overhand loop determines the position of with the tail end inside assuming that the working end is wrapped around the standing part of the rope anticlockwise. Wrapping the standing part with the running end directed clockwise will render the end on the outside as in this picture in which the initial loop was overhand in both instances:

enter image description here

In any event, the typical result achieved in the most common way of tying up the knot is that the tail end of the rope stays inside of the final bowline as in animatedknots.com:

enter image description here

However, starting off with an underhand loop, passing a bight of the standing end through it, and then threading the end of the rope through the bight, results in a nicer final dressing with the tail end on the outside, as in here. This is the final appearance:

enter image description here

This is not even an esthetic issue with the Yosemite bowline, in which the end is parallel to the standing part of the rope.

Are there any reasons to prefer the most typical way of tying a bowline? Mechanically, they seem to be identically sturdy, but the second knot looks less busy and better dressed. Both of them are equally visually verifiable with the "pierced-tongue" final look.

I just came across this:

enter image description here

which would reformulate the question along the lines of:

Is there only tradition behind the "purists" stand?

(*) An underhand loop is formed when the running or working end of the rope is placed under the standing part of the rope. It results in a running loop when a bight of the standing part is pushed through the loop. The opposite (overhand loop) is formed when the running or working end of the rope is placed over the standing part, resulting in slip loop when a bight of the running end is passed through it.

Antoni Parellada
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    It seems to me that the important distinction would be between tail on the inside and tail on the outside. The title of the question talks about overhand versus underhand, but you can change that simply by flipping the knot over. Personally, I and the people I climb with almost never use a bowline for the types of climbing we do. If I want to tie in to a tree as an anchor, I use my cordelette. Using the climbing rope to construct an anchor is kind of old-fashioned and inefficient. Tying in to a gear anchor with the rope is already a hassle, but anchoring to a tree would require untying myself. –  Jun 21 '21 at 23:28
  • https://en.wikipedia.org/wiki/Cowboy_bowline also ABOK 1034 1/2. "In formation the Left-Hand Bowline Knot is similar to the Left-Hand Sheet Bend (# 67). It is often tied directly around a post, in mooring (probably by mistake), instead of tying a Right-Hand Bowline (# 1010), to which it is distinctly inferior." – endolith Jun 22 '21 at 00:31
  • @BenCrowell Very interesting insights. I do use the cordelette for a second anchor point. Any take on the tail end within or outside the knot? It shouldn't make any difference, should it? – Antoni Parellada Jun 22 '21 at 01:03
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    @endolith Well... That is the answer! Thank you! – Antoni Parellada Jun 22 '21 at 01:07
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    @BenCrowell Are you USAian? Building anchors with the rope is very common in the UK. We hardly every use cordelette. Personally, I'd throw a sling around a tree for an anchor but if I didn't have one big enough I'd definitely use the rope. – Darren Jun 22 '21 at 09:54
  • @AntoniParellada OK :) – endolith Jun 22 '21 at 13:05
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    Actually while searching I realized this question may be a duplicate: Bowline knot, end at the inside or outside? – endolith Jun 22 '21 at 13:14
  • @Darren: Yes, I'm Californian, and I'm describing what I think has been normal here for multipitch trad since ca. 1970. It may be a very different story for mountaineering or in other countries. If you anchor to a tree with the rope, do you untie, or do you tie the bowline on a bight? Either option seems extremely undesirable for the kind of climbing I do. For a gear anchor, tying in with the rope is I guess less annoying, but it uses up rope, it doesn't allow much possibility for equalization, and it's very time-consuming if you're not swapping leads. –  Jun 22 '21 at 16:27
  • @BenCrowell none of the above. Rope around tree, back to climber, clove hitched around a carabiner on the rope loop, or maybe the belay loop. – Darren Jun 22 '21 at 16:33
  • @Darren: If you're going to do it with the rope, the clove does sound like a much better option than the bowline. What you're describing does mean that you're going to have more difficulty if certain self-rescue situations come up, since your harness is part of the anchor. –  Jun 22 '21 at 16:36
  • @BenCrowell oh, how would I get the end of the rope round the tree is what you’re asking? Yes, I usually climb trad with half ropes so could untie one if needed, otherwise just walk around the tree, or make a big bight and pass that round. Any number of options really. Sling would always be first choice though. – Darren Jun 22 '21 at 16:38
  • @Darren: Ah, I see. Half ropes make sense. Nobody here climbs with half ropes. Re self-rescue, what I mean is that you can't escape the belay. So do you just never equalize your gear anchors? –  Jun 22 '21 at 16:39
  • @BenCrowell there are ways to escape the system with prusiks and things. – Darren Jun 22 '21 at 16:45

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This is a Cowboy bowline. Wikipedia says:

tests of the different versions' strengths show little difference; conjecture about either knot's vulnerability to some failure remain pretty much only that – conjectures.

But this statement doesn't have any references.

Cordage Institute did a strength testing comparison:

The standard Bowline was selected for the test, but the Cowboy or Dutch Bowline was tested to see if there was a difference. The numbers were almost identical.

One of the references is Ropers Knot Page, which says:

The Dutch Navy uses this variant of the bowline. And, of course, the Dutch sailor says this one is superior. The working end is not so easy pushed back by accident, they say. I think it is just a difference in culture.

It's listed in Ashley Book of Knots as knot #1034½:

In formation the Left-Hand Bowline Knot is similar to the Left-Hand Sheet Bend (# 67). It is often tied directly around a post, in mooring (probably by mistake), instead of tying a Right-Hand Bowline (# 1010), to which it is distinctly inferior.

He doesn't go into any other detail about what makes it "inferior", though.

endolith
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