germany, as addition to nvoigt's answer
Answer for the video linked in a comment (snapshot below) and assuming the siren is actually on (which is plausible but the gif video doesn't have sound, and in Germany the Sonderrechte-magic requires blue flash light and siren)
Are the driver's actions [staying where they are until green, then starting] legal in this situation
NO - they need to immediately make way - unless the fire truck had their siren off. In that case they behaved as they should.
is the driver required by law to enter the intersection on red in order to make way for the emergency vehicle?
Also NO since there is no necessity: they have sufficient space before even violating the stopping line to move sideways and forward so that the fire truck can pass.
(If there is really no other possibility to let the emergency vehicle pass, then yes they may have moved into the crossing)
Details:
Since the car is stopped at the stop line, they cannot move out of the way without entering the intersection
For most cars at most traffic lights this is not true.
(Fully-grown trucks and cars with trailer may have more difficulties, though. Also, e.g. construction sites may restrict the space where one can put the car. But yes, you'd be supposed to fold in your mirror and e.g. move up a tunnel's safety strip "pavement" if that's what is needed)
First, one is supposed to stop well before the holding line and subsequent cars are supposed to leave adequate distance as well. One of the reasons for this is to have maneuvering space to move the car sideways (really squeeze to the side of the lane or go on the sidewalk or embankment) in order to give way for emergency vehicles.
Consider the video: The green car on the 1st lane has (in an exemplary manner) left plenty of space even before the stopping line. They could thus have used it to move far to the right, which would have given the fire truck sufficient space to pass. Yes, the red car on the 2nd lane may have helped as well by moving further to the left.
(BTW, note that the fire truck is labelled in German: Feuerwehr)
Second, intersections with traffic light are on purpose designed with space between the stop line and the actual traffic light which in turn is placed well before the actual intersection begins:
source
Also, the general ban on stopping and parking near crossings means that there are no parked cars keeping you from entering the sidewalk or embankment.
Thus, in case one nevertheless finds oneself without sufficient space before the stop line, there is the additional safety zone even before the actual traffic light.
In Germany, violating the stop line alone is a really minor thing (as long as noone is endangered). Violating the red traffic light happens when crossing the actual post with the signal. And that is still well before entering the crossing. (Side note: automatic traffic light violation cameras usually record several levels of violation, and distinguish stopping line violation from red light violation/actually entering the crossing. In the situation discussed here, the timestamps would also show that the violation did not happen at speed but by a very slow car, plus possibly the position of vehicles indicating that they form a passing lane)
Since fines need to be proportionate to the offense, the catalogue of traffic fines at the KBA may be used as convenient surrogate to compare the seriousness of different violations.
Not giving way for an emergency vehicle with flashing blue light + siren on is above red traffic light violations that did not actually endanger someone.
138600 Sie unterließen es, einem Einsatzfahrzeug mit eingeschaltetem blauen Blinklicht und Einsatzhorn sofort freie Bahn zu schaffen.
You failed to immediately give way to an emergency vehicle with its blue flashing lights and siren switched on.
§ 38 Abs. 1, § 49 StVO; § 24 Abs. 1, 3 Nr. 5, § 25 StVG; 135 BKat; § 4 Abs. 1 BKatV
category A; 2 points + 240,00 € fine + 1 month without driver's license
141211 Sie hielten nicht an der Haltlinie (Zeichen 294).
You did not stop at the stopping line (sign 294)
§ 41 Abs. 1 iVm Anlage 2, § 49 StVO; § 24 Abs. 1, 3 Nr. 5 StVG;
154 BKat
10,00 € fine
137600 Sie missachteten das Rotlicht der Lichtzeichenanlage.
You disregarded the red light at the traffic lights.
§ 37 Abs. 2, § 49 StVO; § 24 Abs. 1, 3 Nr. 5 StVG; 132 BKat
category A; 1 point + 90,00 € fine
137618 Sie missachteten das Rotlicht der Lichtzeichenanlage. Die Rotphase
dauerte bereits länger als 1 Sekunde an.
You disregarded the red light at the traffic lights. The red phase
lasted longer than 1 second.
§ 37 Abs. 2, § 49 StVO; § 24 Abs. 1, 3 Nr. 5, § 25 StVG; 132.3 BKat;
§ 4 Abs. 1 BKatV
category A; 2 points + 200,00 € fine + 1 month without driver's license
for comparison, this one is triggered even without any emergency vehicle being around, as soon as the traffic slows down to walking speed (but this happens far from a traffic light):
111600 Sie bildeten auf einer Autobahn oder Außerortsstraße keine
vorschriftsmäßige *) Gasse zur Durchfahrt von Polizei- oder
Hilfsfahrzeugen, obwohl der Verkehr stockte.
You did not form a lane on a highway or out-of-town road
lane for the passage of police or emergency vehicles, even though traffic was at a stagnated.
§ 11 Abs. 2, § 49 StVO; § 24 Abs. 1, 3 Nr. 5, § 25 StVG; 50 BKat;
§ 4 Abs. 1 BKatV
category A; 2 points + 200,00 € fine + 1 month without driver's license
There are additional violations with increasing severity in case someone was endangered or the violation actually having caused an accident (for all of them).
Some legal opinions saying that red light may be violated in order to allow the emergency vehicle to pass as long as noone is endangered. Nevertheless, one should not go further than absolutely necessary (All in German):
https://www.gansel-rechtsanwaelte.de/schlagzeile/was-darf-man-um-dem-rettungswagen-platz-zu-machen, translation:
If an ambulance is approaching from behind and you have no possibility of driving to the side due to a narrow road or roadworks, you may even drive through a red light. Please note that you are not allowed to cross the entire intersection, but only the stop line. This must also be done in such a way that no other road users are endangered. So drive slowly up to the red light, watch out for crossing traffic and position your vehicle at the junction so that the ambulance can pass.
If a camera was installed in the traffic light system that was triggered by your maneuver, you are usually off the hook. As soon as you hear a blue flashing light in combination with a siren, all vehicles must make way! If you receive a fine notice some time later, you can state that you swerved to avoid an ambulance. Ideally, you should be able to provide the registration number of the emergency vehicle, organization, time and date so that the fining authority can check what happened. However, the proceedings against you should generally be discontinued - nor can you be required to pay a fine.
https://www.anwalt.de/rechtstipps/rettungswagen-im-einsatz-ueberholen-davonrasen-und-andere-verkehrsverstoesse-erlaubt_159835.html, translation:
What happens if you are at a traffic light, an ambulance comes up behind you and you have no room to pull over? Can you just drive through the red light? The most important basic rule in this situation is that you must not endanger other road users!
In principle, you are allowed to drive through the traffic lights in this case. However, it is advisable to only drive as far as necessary over the stop line and not necessarily into the intersection. Once you have let the emergency vehicle through, it is best to wait behind the traffic lights for the next green phase or, if possible, drive back behind the stop line.
All translations by deepl with some manual help by me