Although the below government guidance confirms that I don't need to make a new application, does my sponsor need to notify UKVI of the change of the job title from 'Teacher of A and B' to 'Teacher of A and C'?
Or it is not considered a change of job because it is the same title of 'Teacher' anyway?
I kind of not want my employer to report it as the HR lady familiar with visa has left, and the new HR specialist knows absolutely nothing about it. If I do not tell her about this, she would for sure not know! But I am worried reminding her about this would open another can of worms

Also it is the sponsor's responsibility to inform the UKVI not mine. If by rules my sponsor needs to notify UKVI of this minor change and they did not, will this have any impact on my future Skilled Worker Visa and Permanent Residency applications?
A person does not need to make a change of employment application if:
• they are staying with the same employer and changing their job to one in the same occupation code, and the change does not mean moving from a job included in Appendix Shortage Occupation List to one which is not included
• their basic pay increases
• they are moving under either:
o Transfer of Undertakings (Protection of Employment) (TUPE) arrangements
o equivalent statutory transfer schemes
o the Cabinet Office Statement of Practice on Staff Transfers in the Public Sector,
they are still working in the same occupation code and they would still score 70 points if they were to make a new Skilled Worker application
• their pay was reduced or stopped during a period of absence of less than 4 weeks in a calendar year or (if longer than 4 weeks in a calendar year) for a reason listed in paragraph 9.30.1. of Part 9 of the Immigration Rules.
• their salary is reduced, but they would still score 70 points if they were to make a new Skilled Worker application under the Rules currently in place
Sponsors must still notify UKVI of any of the above changes (other than basic pay increases due to annual increments or temporary salary reductions permitted under the Sponsor Guidance).