Beginner question: a 4-4 enclosure requires two stones to ensure that the surrounded territory is safe. This shows the position of the second stone with a knight’s move:
How about a one point jump? What’s the position of the second stone?
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My rough impression would be that black making an additional “enclosure” move in the one point jump case at all is a bit more unusual because it’s a bit less important. It’s less important because unlike sealing off the corner after the knight’s move where the corner does become territory (or almost does), in the case of the one point jump it still leaves the top open from the fourth line, so it’s not doing as good a job at sealing territory.
Instead, with the one point jump as black I would want to first build outward, making varyingly wide extensions on the top side or right side, and/or further building those extensions to be center moyo if those side extensions are already in place, before coming back to the corner. And depending on where and how white invades, and what fights spill out, you’ll get different situations with different nearby stones and so the question of what enclosure to use when it’s just all empty with no additional white or black stones nearby tends not to come up in the first place.
If we insist on the case where it does come up that black wants to add a stone the corner and there’s nothing too nearby, it might to be the kind of situation might be where black has already made extensions on one or both sides from the corner, and/or already has some kind of moyo spanning the whole top quadrant of the board if not more. And white has not gotten time to invade (but presumably made up for it elsewhere on the board). And maybe now that black is adding yet another move, black wants to play an ultra-solid move to consolidate the moyo and deny white good invasion spots to gain a foothold in the moyo. In that case, I would imagine the iron pillar (3-4 point to the right) or even directly playing the 3-3 point. Such moves are very slow, but if you’ve run out of efficient ways to expand and still have yet another free move to consolidate, then they leave very little aji for the opponent to play with for invading. Depending what the sides look like, I could also imagine the same diagonal move (3-5) to also be occasionally good.
Could be wrong, this is just my intuitive feeling (weak amateur dan player). No idea what the bots say since they tend not to develop the 4-4 point with a one point jump in the first place, so it also doesn’t come up in bot games either.
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So - to me - this is a bit of a trick question, the reason being that both of the examples you noted above are vulnerable to a 3-3 invasion, and thus not securely enclosed
As I understand it, your first example can be invaded like this
your second example can be invaded like this
The way I learned it, you can enclose a corner with two stones by playing the 3-4 and then a knight’s move from that
However, the 4-4 is difficult to enclose securely because the 3-3 offers so many options for invasion
Not sure how often you’d encounter these variations out in the world, and it would probably depend a lot on the skill level of your opponent, but yeah, the potential for invasion is there
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Darn I thought I was going to learn a new trick - how to make a 4 - 4 enclosure with 2 stones 
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I’ve seen players do this
Black can still get most of that corner even when approached, but it’s not enclosed per se, just shooing away that pesky 3-3
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I think White may be dead in that corner unless they can pull of some Ko magic
And yes, I think your 2nd example kills White as well
Hmmm - maybe you CAN enclose a corner from a 4-4 
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Both of those are bad sequences with several mistakes. First one 4-4 plus knight’s move and kosumi does secure the corner. Black should play 4 at 5 if he wants corner. 4 on outside is also possible if black decides outside is more important, but it’s black’s decision to give up the corner. White 7 at 8 lives without 2nd line crawl which might die anyway depending on outside (when this shimari is made black will often have an extension on that side).
One point jump indeed doesn’t secure corner, but 3 and 5 playing into already defended hanes is terrible beginner shape mistake. Kosumi q18 is better, or s16 and aim ko. But going back further though the 3-3 isn’t secured, against that enclosure 3-4 attach is usually a better invasion and then black has choice to keep corner and you make group on side or keep side and you get roomier corner.
Cunningham’s Law eh?
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Big mistake from black (without nearby support) due to outside cut after. Original sequence was already great for black who got thick seal in sente. Don’t interrupt your opponent when they are making a mistake.
Indeed variations may or may not work depending on surrounding stones, and cuts may be problematic if white stones are around.
Even though this shape doesn’t fully secure the corner territory, it’s usually still called an enclosure, specifically a one-space enclosure: 4464 enclosure at Sensei's Library
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And even more specifically a one-space high enclosure.
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I’m beginning to wonder what is even the point of 4-4 if the 3-3 invasion can’t be refuted. It feels like trying to take a corner without really taking it. I gather there’s an Idea of influence towards the center but if I WANT that corner should I just start with 3-3 or 3-4?
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When you follow it up with a small knight’s enclosure, the corner territory is fairly secure (they can still get a ko, but usually it’s not so good for them to do that in the early game). So you can still choose a territorial strategy from 4-4.
That being said, 3-4 focuses more simply/strongly on a territorial strategy. Traditionally 4-4 is aiming more at speedy/flexible development and influence, and less on secure territory.
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You describe it very good I think.
4-4 is very flexible. It takes the influence if invaded, takes a bit of both in case of some popular Joseki, or takes the corner if the game goes in a way the corner gets interesting.
If you want pure territorry from begin with, yep, 3-3 and 3-4 are much better options.
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3-3 and 3-4 are definitely better at claiming the corner than 4-4, so you may enjoy playing them.
However (in my opinion), it is better to approach a game of go with a flexible mindset. Instead of deciding on a fixed plan from the very beginning (like “I claim this corner” etc.), it is better to react to your opponents play in a way that is as beneficial to you as possible, given the whole board situation.
I think that 4-4 as an opening move incorporates this idea very well, and strikes a unique balance between having the potential for creating corner territory and influence towards the center.
Ultimately it’s a matter of style though, and especially for learning, it’s good to try out all kinds of things.
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Thanks for the answers. To give a little more context, my question arose while reading the second volume of the Learn to Play Go series, where you can find this:
There’s some strong phrasing there with “an additional stone … to secure the corner territory” and the same with the 1 point jump enclosure, so I interpreted it as a configuration that really ensures a big chunk of the corner for Black. Ps: The comparison at the beginning is with the 3-4 point where she mentions that only one knight move is enough to enclose the corner, as @tonybe said.
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If you’re interested there is a book (by Shuko the famous one) named “enclosure josekis” which is in English and is only about this.
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Thanks for the suggestion! But would that answer my question? Since a joseki involves two players alternating moves, even if I find the josekis where the patterns I posted appear, wouldn’t the order of the moves change everything, since my question is about the existence of a “static” pattern that secures the corner. A little bit less useful, but it’s more a curiosity.