This is something I have thought of in case I ever need to provide an injection. I have given injections to many humans and my personal cat. The type of medication, the route of delivery, and the tool being used all need to match up. If the medication being delivered should be administered under the skin then the syringe components used need to be appropriate to complete that task. There are also industry size standards which limit any ideal injection delivery method when it comes to non-human patients.
For the syringe the most likely scenario is a 1 mL plunger with a 1.5 inch length single regular bevel needle. The needles come in different gauges so I would have to defer to a veterinarian on what gauge to use. Some medications are more or less viscous than others. The video had a shorter needle length, maybe 0.75 inches, the gauge looked small around 25. The smaller the gauge the more force is needed depressing the plunger.
Needle bevels have a tip and a heel for cutting edges. The tip is the outermost point. The heel is where the needle opening meets the shaft of the needle. Therefore the needle opening is between the tip and heel, called the lumen. The lumen should be facing the person giving the injection. This is for a couple reasons. One is that both the tip and heel are cutting edges, only have one cutting edge against the skin of the patient is to prevent any additional skin being cut beyond the point the tip has punctured. Second, it is also visual to see when the lumen and heel of the bevel has completely gone under the skin, or if the lumen is visible then the needle must be pushed forward to be under the skin again. The angel of entry for a human subcutaneous injection is 45°, but my best guess for a bearded dragon is 10-15° angle which is considered an intradermal injection for humans. Using a 10-15° angle, which matches closely to the video, would likely result in a subcutaneous distribution of medication for bearded dragons.
There are also finesse points to giving an injection. Typically there is a slight force when inserting the tip of the needle followed by a very slight pause, when aiming for a subcutaneous injection. The pause allows for skin to snap back to shape so that needle is not inserted further than needed, if the lumen or heel are still visible more force to insert is required. Depending on the medication and skin of the patient once the lumen is completely under the skin the needle may need to be pushed in further then retracted, thus creating a pocket of space for the medication. The needle could be pivoted with the entry point of the skin being the central axis of the pivot, keep the 10-15° angle, again this is to create a pocket for distribution. The angle of the needle could be brought down to 0° which is to create a pocket under the skin for medication to be distributed. These pocket creating techniques can be combined, this is where experience comes in and if you can feel by hand that there is a pocket, this has to be learned by experience. In any scenario return the needle to a 10-15° before pushing down the plunger. Having the medication back up and exit the entry point of the skin is to be avoided which is why all the trouble of creating a pocket is needed. Then you have to do all the above very quickly when a patient is resisting, which I assume most dragons would be doing.
The reason I wrote all that above was to give more explanation for what was happening in that video. The spot on the dragon by the shoulder is a good location. I would choose that spot. The bone structure beneath the skin helps prevent the needle from being inserted too far, the bones would provide great resistance, as in it would take great force to insert past the bones, which is to be avoided. Alternating left and right for repeat injections is also appropriate. She was able to pull up on the skin, from the pinched portion she made a tent with an indent for insertion, due to the size of the patient an indent formed by the finger in the skin tent which she aimed for. She said you will feel a “pop” as the needle breeches the dermis through to the subcutaneous layers. (One thing to note with the tenting method of the skin, if you do not pause after the pressure of insertion you may breech the other side of the tent, and if someone did, release the tent to see if the skin snapping back to place allows the lumen of the needle to be under the skin again between where the insertion and the accidental exit point was) Then she let go of the skin tent. This allowed the skin to snap back into place. She pulled the needle back slightly after releasing the tent. Both of those actions, release the tent and pullback of the needle, create a pocket in the subcutaneous layers. I keep saying pocket, any membranes holding the layers together need to be separated by mechanical trauma of needle insertion. Any small blood vessels presented are ruptured which prevents the very small chance of an accidental intravenous injection. She mentioned as much in the video this helps avoid blood vessels, it destroys the blood vessels but the point is to avoid intravenous injection so the goal is achieved. The video It appeared she had a 10-15° angle of insertion, overcorrecting by telling people to try for a parallel insertion is how your average person injecting their animal at home, with no prior experience, has a chance to reach that 10-15° angle.
Overall that is a good video to get people to do things right without thinking of all the mechanics involved. Towel to assist holding the bearded dragon. Make the experience as positive as possible with some kind of positive reward. The more positive the better. She had basic points like turn the syringe with the needle up to plunge out excess air. The medication temperature, this could be dependent on the medication, I would defer to a vet what temperature the medication should be. Any injection video I have seen for reptiles has the needle being inserted against the growth of scales, to aim under the scale as the video stated. An uncooperative patient with scales I could see being a risk if the patient struggled, and your needle aim was in line with the growth direction of scale, the needle could miss and start to slide over the scales which is dangerous for the one giving the injection and the patient. Going against the scale growth would help one aim below a scale to the skin. I may not be explaining well, that resistance against the growth direction under a scale would be noticeable and an assuring sign that the syringe was properly lined up to enter the skin. I have not injected any scaled patient, but that is what I expect the experience would be like.
Rubbing the spot of the injection gently afterwards helps distribute medication and may also calm your dragon down. The needle injection is sharp so it feels sharp, where a finger has a larger area of pressure. Stimulating more nerves in a gentler manner than the injection lets your dragon know there is no more sharp feeling. This technique works very well for people too.
Had to edit for some typo. *