How to merge or move two vertices without affecting the others?

I want to merge two vertices into one, but Blender help me to select the third one automatically. I didn’t turn one Proportional Editing Mode. Would anyone here can explain why? Please go to my shared videos below to see my problem in detail, if my word is not clear enough. Thank you.

No. They can only guess at best. No example file to check for other options, so my only guess is that automatic selection method in user preferences -> system chooses a problematic one. Switch to one or the other.

Look at the header. You have 4 vertices selected, not two.

select the vertices manually using SHIFT + RIGHT-CLICK (to select multiple) make sure you’re in edit mode and press ALT+M (this will bring up a merge menu) select either “at first” or “at last”. this way requires a little bit of manual labor, but it should fix your problem.
best of luck,