Nah, this would be the labor force participation rate - and that one took a plunge since 2008 and never recovered since.
The OECD chart linked (if you play with the sliders) shows that both are increasing globally. I know that in the UK the employment rate is at an all-time high [3].
Your link seems like it might be using a strange measure of employment [4] which excludes agricultural workers and the military in the numerator and includes all people who are aged 16+ in the denominator, so as your population ages, the indicator would worsen.
[1] https://data.oecd.org/emp/employment-rate.htm
[2] https://data.oecd.org/emp/labour-force-participation-rate.ht...
[3] https://www.ons.gov.uk/employmentandlabourmarket/peopleinwor...
[4] https://www.investopedia.com/terms/c/civilian-labor-force.as...
I think you might be confusing cyclical and structural unemployment with your first point. I think it is valid to talk about the two numbers separately as they deal with pretty different problems (labour market liquidity/efficiency versus skills shortages).
The employment rate is at an all time high in many countries.
As for point 2: employment in the black economy is usually not included in official employment figures; people who want to work more hours are classed as underemployed although data on this is not as comprehensive as for the other measures.
I do not think the definition of employment has changed much in the last decade. Do you have a source?