TL;DR - linguistically Mandarin is equivalent to 官话, but be careful when using the concept with Chinese people. 北方话 is more widely understood, but even then... caveats abound. "Standard Chinese" is equivalent to the standards embodied by 普通话／国语／华语, and "Standard Mandarin" is an alternative name for this in the West, but this term has no cognate or calque in the Sinosphere (note that non-Chinese Eastern perspectives are different again).
There are several issues here:
- language vs 语言 vs dialect vs 方言 / topolect
- Mandarin vs Standard Mandarin vs 普通话／国语／华语 vs 官话 vs 北方话
- approach to speech varieties: labelling by social construct (Mandarin, 官话 and 普通话／国语) vs by locality (北方话) vs by classification through linguistic features (also "Mandarin", but also "Yue" 粤语).
I would recommend doing some research into each of these separately, and to see how popular perceptions are in China and outside, and how linguistic research uses terms carefully to describe each of these aspects. A good bit of research into the history of Standard Chinese would be of use to you.
The level of linguistic knowledge required by someone working in northeastern Fujian to know that the topolect of Fuqing 福清话 and that of Fuzhou 福州话 are closely related won't require that person to be aware that this is classified into the Min Dong 闽东话 family. Similarly with the situation in the north; a Northeasterner 东北人 from Shenyang 沈阳 might have had to adopt speech patterns and listening skills to work in Jinan 济南, Shandong, but knowledge that these two are separate branches of sensu stricto "Mandarin" 官话 would not be necessary. That person would acknowledge that both were 北方话, though.
These linguistic branches are "not really a thing" in the popular culture of China, whereas "Standard Mandarin" is in the (intellectual) life of the West, as it is taught as such in Western classes of standard Chinese. It is similar to the 字 vs 词 issue: these labels do not map to the same linguistic reality as letter vs word.