We have three main sources that tell us about the use of medieval swords. First, there are surviving instruction manuals for fighting with medieval swords. Second, there are artistic depictions of fighting. Third, there are the weapons themselves.

The fight manuals are from relatively late in the medieval period. The oldest is the manuscript known as Royal Armouries Ms. I.33, from about 1300, covering the use of the one-handed sword with buckler (i.e., small shield). Later fight books cover a range of weapons, including the longsword (AKA hand-and-a-half sword), various one-handed swords, spears, halberds, pollaxes, and large two-handed swords. A variety of fight books can be read at http://wiktenauer.com/ which aims to make these works, and translations thereof, available online. From these, we know a range of techniques that were used in fighting.

Artwork showing fights also tells us about fighting with swords (and by this, I mean art other than the illustrations in the fight books). It typically isn't technically-oriented like the fight books, but it tells us about what kind of things the writers and readers of the manuscripts where the art appears thought would represent sword fighting visually.

From the combination of these two sources of information, we expects that, typically, a mix of cuts and thrusts would be used. For example, both are shown in use here:

https://commons.wikimedia.org/wiki/File:Rochefoucauld_Grail.jpg

Fighting was not focussed on lunges and thrusts, although thrusts were an integral part of fighting. (The lunge seems to be post-medieval, appearing in fight books from the 16th century, when the swords in use were no longer the medieval "broadsword". There are attacks delivered while stepping forward, either a step led by the front foot (as used in modern fencing) or a step with the rear foot (which then becomes the front foot), but these are not the lunge as we know it today.)

Thrusts are more important in armoured fighting, but are, with the longsword, often delivered while "half-swording", holding the sword with one hand on the hilt and the other half-way up the blade, as shown here: https://commons.wikimedia.org/wiki/File:Augsburg_Cod.I.6.4%C2%BA.2_(Codex_Wallerstein)_107v.jpg

For a demonstration of armoured fighting with the longsword, with prominent use of half-swording, see https://www.youtube.com/watch?v=1S_Q3CGqZmg

For unarmoured fighting, a mixture of cuts and thrusts is useful. These can be used alternately, for example with a thrust intended as a feint followed by disengagement from the parry into a cut, or a cut intended as a feint followed by a cut. In particular, thrusts tend to come in directly from the front, while cuts come from the side. Thus, a cut can be used a feint to pull the defence to one side of the body, clearing the space in the centre for the follow-up thrust. These kinds of techniques are useful whether the defence is with a shield, or with a sword held in both hands. Both cuts and thrusts are also used with a one-handed sword when used without a shield. From Lecküchner's work on fighting with the messer (a single-edged sword),

• cut: http://wiktenauer.com/wiki/File:Cgm_582_004r.jpg (right is cutting left's wrist)

• thrust: http://wiktenauer.com/wiki/File:Cgm_582_004v.jpg (left is thrusting at right's chest)

The fighting in Kingdom of Heaven (your second clip) is grounded in historical technique. The high guard was used, and cuts similar to those shown in the clip were used from that position. More thrusts would have been likely. However, the combination of thrust and cut (e.g., from high guard, a cut pulling to one side and coming in diagonally or horizontally followed by pulling the blade back to avoid contact with the block/parry and to get onto the other side of it, and coming in with a thrust) takes training and development of skill to become natural enough to use in the stress of a fight, and it would be no surprise to see a fighter simply using a rain of successive cuts.

For typical movie/TV choreography, there are distortions in the fighting. The action is slowed, and made more obvious to the viewer. This "more obvious" is done by (a) making movements bigger and slower and (b) making less use of deception. If there are multiple opponents, they will often attack in an orderly sequence rather than together. While a mix of cut and thrust might be more realistic, cuts are often favoured in fight choreography. First, it's easier to make them big and showy (and slow), and second, cuts also make for safer fight choreography. Often, there are major departures from reality, such as armour becoming easily cut through (at least by the heroes), complete ignoring of teamwork in battle scenes, battle scenes with infantry with no armour and no shields and almost no weapons other than swords (unlike the spear/polearm and missile dominated real-world battlefield) - first clip, I'm looking at you! But the use of many cutting techniques, including as the dominant sword techniques, is realistic. Wild frenzied hack-and-slash, while not the best fighting technique, can certainly emerge in the stress of battle (Kurosawa often used excellent fight choreography, and on multiple occasions showed the difference between skilled and unskilled fighting, and the effects of extreme stress on fighters. His movie The Seven Samurai is a superb example of this, showing a variety of realistic fights and fighting techniques. This isn't medieval European fighting, but there are many, many similarities between Japanese and European fighting technique (e.g., similar cuts, similar thrusts, similar guards).)

For an example of movie-style fight choreography using realistic medieval technique, see https://www.youtube.com/watch?v=Cn36Pb8z3yI

For a look at sword and buckler technique, see https://www.youtube.com/watch?v=QQAjrjlBaL8 and https://www.youtube.com/watch?v=XU9SKMDFtm4