What is "背心" called in English? Photo attached.

These are called tank tops, sleeveless shirts, or less commonly, wife beaters

Unfortunately this is highly region-specific. You'd be better off asking a native speaker from your particular target audience.

In the UK, if they are designed to be worn under other clothes then they are a vest. If they can be worn just on their own, they are usually either a vest top or a sleevless tshirt.

The first image is of a tank top, or less commonly a wifebeater. It is an undergarment, sometimes worn as a shirt. The second image is of a vest, usually worn over a shirt, akin to a sweater but not covering the arms.

I would call both a vest.

In the US these are not the same thing. Top photo is a tank top, and bottom photo is a sweater vest.

In Australia the first one is a singlet, tanktop or wifebeater. The second is a sleeveless top, sometimes a singlet. The difference is whether it is tshirt quality (sleeveless) or thin and designed to soak up sweat (singlet).

This varies tremendously by region!

The formal term for that is an "A-shirt," because it looks like an "A," as compared to a "T-shirt," which covers the shoulders and some of the upper arms and looks like a "T." Native speakers routinely use the term "T-shirt" or "tee," but they rarely use "A-shirt," you see that term online when ordering them. The informal term is "tank top" or "sleeveless shirt," although the most common term is "wifebeater." But "wifebeater" should generally not be used because it is offensive on multiple levels: not just the reference to domestic violence, but also because of the implication from it that it is worn as a shirt by people in poverty (instead of as an undershirt) and they are more prone to domestic violence.

It's a tank top. The slang is wife-beaters because the stereotype of people who wear tank tops beating their wife. If the shoulder part is thicker, in my mind it's more of a sleeveless shirt or a muscle shirt

If it’s worn by a woman: tank top. If it’s white and worn by a man: it can be called an undershirt, or a wife beater (which is a very unflattering term because it implies the man hits women). Essentially If it’s any colour and worn by a man: it’s called an undershirt but white has its own special name.

Singlet.

In British English yes it's a vest. As pictured the other is a a pullover (despite the no sleeves) or just a "v-neck", if it was the same but a button up it could maybe be called a cardigan (again despite the no sleeves but that's not normally seen) but it could be the same as a suit version which is a waistcoat. In American English it's a tank/Tank-Top (women's ones are pretty exclusively called this), but the mens ones are sometimes called an undershirt (if made to worn under a dress shirt) - note you might hear it called a wife beater as well (generally just the white ones) but I would not recommend using it was it's use is viewed negatively more and more (but used enough you should know what it means).

Vest, or skivvy.

Vest is the UK name. 

In the UK those are vests though the second picture looks like a tank top which is an overgarment not an undergarment. We don't use the phrase wife beater in the UK though I imagine most people would know what it means.

id call the first photo a tanktop and the second a sweater vest

Tank top for both. A vest can look similar to the second image but has a zipper or buttons as it is designed to go over your t-shirt when the weather is a little cool but not wet or snowy. Most vests also have long sleeves but vests for servers or in professional settings may look similar but in black.

A-shirt or undershirt

For men - a vest. Women - singlet / (half) slip.

The third picture (same as the first) is a picture of four tank tops. The second picture is a vest. The difference is that tanks tops tend to be lighter material and/or undershirts, whereas vests tend to be heavier material and/or worn over another shirt.

Top is a tank top or wife beater. Bottom is a sleeveless Tshirt.

The top picture with the four items pictured are called tank tops. The bottom one most closely resembles a vest, but it seems like a thinner material than what they’re normally made of.

It's interesting because I'd have called the red shirt a "muscle shirt" and the top picture a tank top