Most melodies - in any style - can be transcribed using: 1) pitches, 2) rhythms, 3) motives, 4) phrases, 5) overall form.
Unfortunately, in the videos you cited, the musician does not play the rhythms consistently each time the example is played. Therefore, I based my transcription on the overall performance of the second video (the one where the keyboard is set to synthesize a flute).
1) Identify the pitches in each motif (short musical kernel). Note that in these motifs I disagree with the analysis given on the page/video you referenced. My analysis is as follows:
2) Count out the rhythm identifying not only the articulations/attacks, but also the durations, the groupings, and the accents (louder attacks). Developing this skill requires more instruction than can be gained on a web page. However it is a skill that is gained fastest by simply doing transcriptions.
Count and diagram a few possibilities before choosing the best one. The grouping of the pitches over a prolonged passage will indicate the underlying meter, then time signature.
The accented beats, the harmonic motion (there is none in the videos), and the resolution may indicate where the strong beats (or beat number one) occur in each measure.
Shift the motif to a different starting beat (this one starts on beat four) until you start to see the pattern, especially a recurring pattern throughout the entire piece/movement.
When you are confident of the rhythmic pattern, clean up the notation to match the conventions that you have learned. Ask a friend to suggest the correct notational markings.
If there are two possible correct ways to notate a passage, usually the one that uses the least ink on the page is correct.
3) Repeat that process for each motive. Fortunately, you will find a surprising amount of repetition in music. Therefore once you have the first motive done, the next will come to you easier.
4) The motives create phrases. These longer lines will (metaphorically) make statements, require responses, perhaps repeat themselves, perhaps lead to another place or type of phrase. This element of storytelling without words is key to good melodic composition. Once again, after the first phrase is transcribed, the next ones will come more easily.
Here is a raw transcription of the melody you chose, organized into phrases:
5) Analyse the overall form. Observe where the phrases repeat, and how to make the transcription easier to read.
This is the time to add the key signature (one flat), articulations (see the last 2 measures, and make sure that the form agrees with the conventions of the desired musical style. Make sure you indicate the title, so your musicians know what to call the piece. Now is the time to take a quick look online for the composers name - give the author due credit. And you might as well give yourself a little credit too, for your work on the transcription.
Here is the completed transcription:
It gets easier to do with each transcription! Work hard, and good luck!