As you are probably aware, the English language is not always clear cut.
I can see the confusions you are talking about because as a British person, whilst trying to work out a way to answer the question I came up with some ways in which British English (BrE) and American English (AmE) is interchangable in Britain so it is not straight forward.
In the context you are talking about (studies), as you have seen from the website you linked to, class can refer to
- a group of students (countable),
the site indicates class is used in this context in both Bre and AmE
- a teaching period (which can be countable and uncountable)
the site indicates class is used in this context in AmE but lesson is used instead in BrE
- a period of study (countable),
the site indicates class is used in this context in AmE but course is used instead in BrE
Because of the higher proliferation of AmE being heard and read within Britain compared to the past due to access to American movies (movies being a word also used extensively in BrE as well as the relatively excusive BrE films as an example), there is a lot of interchangeability of words between AmE and BrE within Britain, however some AmE words are not always used within conversation between two BrE people.
In general BrE speech
Group of students
Class is used to refer to a group of students in a teaching environment such as a classroom, however, outside of the classroom environment, this group of students would be referred to as a study group.
Teaching period
Generally speaking, this would be a lesson in BrE.
No talking during the lesson
However, there are times when class is used
No talking in class
I would say the key part of this is when during is used. During class is not generally used in BrE although it would not be unexpected or misunderstood if it was used.
On the question of class or lesson being used in BrE for Bible class in AmE, as far as I am aware, within BrE it is usually Bible lesson or to make things more confusing, Bible study is also used.
As for after class or in your example, after lesson, in BrE it can be either
I will see you after class
or
I will see you after the lesson.
To add to this, you can also use
I will see you after the class/classroom session
or
I will see you after the study session
or
I will see you after study
Period of study
In the website, this is referring to a period where several class sessions are involved.
Course is definitely used in BrE and if you was to refer to a series of classroom or lecture room sessions as one class, this would cause serious confusion here in Britain.
I am a little confused with this as I have not personally heard anyone use the word class to refer to a series of classroom sessions unless they are saying something like
We only have five classes to go 'til the end of the course.
Maybe someone within the AmE community can help with this.
In the context of your example, in the UK* we would typically use the word Year, e.g.:
I am in Year 11
(Though, Year 11 in the UK would be roughly equivalent to 10th Grade in the US)
Grade would typically be used to describe the results of an exam.
*more specifically, in England, since education in the UK is devolved to each of the home countries' respective governments.
Best Answer
There is overlap in the meanings and it depends, to some extent, on whether you are using AmE or BrE.
In the UK a class or form tends to mean the same thing. A group of young people who study in the same classroom during one school year. In most schools there would be several such forms who, together make up a year. Most UK schools so not use grade in this context, reserving it for exam results ranking.
In the USA I believe class can mean either a single group, as in the UK, or the entire year, as in "class of 2019". Grade is used to distinguish the year groups. First Grade being the entry level for the youngest children, often known as "reception"or "Year 1" in the UK. I'm not sure that the Americans use form that much, but if they do it will probably be the same as in the UK.