The Berne Copyright Convernmtion is adhered to by almost every country oin earth, including all EU member countries, the UK, and the US.
Articles 9 and 10 of the convention provide that:
Article 9 - Right of Reproduction:
(1. Generally; 2. Possible exceptions; 3. Sound and visual recordings)
(1) Authors of literary and artistic works protected by this Convention shall have the exclusive right of authorizing the reproduction of these works, in any manner or form.
(2) It shall be a matter for legislation in the countries of the Union to permit the reproduction of such works in certain special cases, provided that such reproduction does not conflict with a normal exploitation of the work and does not unreasonably prejudice the legitimate interests of the author.
(3) Any sound or visual recording shall be considered as a reproduction for the purposes of this Convention.
Article 10 - Certain Free Uses of Works:
(1. Quotations; 2. Illustrations for teaching; 3. Indication of source and author)
(1) It shall be permissible to make quotations from a work which has already been lawfully made available to the public, provided that their making is compatible with fair practice, and their extent does not exceed that justified by the purpose, including quotations from newspaper articles and periodicals in the form of press summaries.
(2) It shall be a matter for legislation in the countries of the Union, and for special agreements existing or to be concluded between them, to permit the utilization, to the extent justified by the purpose, of literary or artistic works by way of illustration in publications, broadcasts or sound or visual recordings for teaching, provided such utilization is compatible with fair practice.
(3) Where use is made of works in accordance with the preceding paragraphs of this Article, mention shall be made of the source, and of the name of the author if it appears thereon.
Article 10 section (1) permits quotations from public works, within limits, and Article 9 (2) and 10 (2) permit the laws of individual countries to grant more specific exceptions.
Consider as an example the copyright laws of France. Article L 122-5 (in WIPO's translation) provides that:
122-5. Once a work has been disclosed, the author may not prohibit:
private and gratuitous performances carried out exclusively within the family circle;
copies or reproductions reserved strictly for the private use of the copier and not intended for collective use, with the exception of copies of works of art to be used for purposes identical with those for which the original work was created and copies of software other than backup copies made in accordance with paragraph II of Article L. 122-6-1;
on condition that the name of the author and the source are clearly stated:
(a) analyses and short quotations justified by the critical, polemic, educational, scientific or informatory nature of the work in which they are incorporated;
122-5-2 seems to cover private notes, and 122-5-3 (a) seems to cover critical commentary that is published. Between them thes seem to cover the cses in the question.
The question also asks:
I would also like to understand what laws are actually applicable in this case. The laws of the country of the copyright holder/publisher? The laws of the country where P resides? The laws of the country where the content is hosted? The laws of the country/countries where the content is available?
Unfortunately, the only possible answer is "any or all of them". The copyright owner can choose to sue in any country where an act of infringement has occurred. What counts as such an act and will permit suit depends on the law of the particular country, but it is likely that suit can be brought in any of the countries mentioned in the question. Once suit is brought, the laws of the country where it is brought will generally apply. If the plaintiff wins and obtains a judgement, s/he may apply in other countries to have the judgement enforced there, and may often obtain suh enforcement.
Note however that it is a time-consuming and costly business to bring a copyright suit, much less then have it enforced in a different country. Most copyright owners will not bring such suits unless they think it likely that they will recover sufficient damages to make this profitable. But that is not a rule of law, and some owners may choose to persist in unprofitable suits, perhaps to deter other infringers.