Learn English – In the phrasal verb ‘put up’, what is the meaning of ‘up’

I think the idiom you are looking for is to take a backseat, which means to be in a position of less importance or a position of not being in control. If you’re in the driver’s seat, you’re in control of the car; if you’re in the back seat, you’re not in control.

Here are some examples of this idiom from the Corpus of Contemporary American English (COCA):

“The sandwich was gooey from cheese and crisp from the grill, though the ham took a backseat to the other ingredients.”
“Search and rescue had taken a backseat to all the duties of a sheriff to serve and protect”
“Problems of the developing world, which have often taken a backseat at G-8 summits, look likely to be pushed to the sidelines yet again”
“But questions like these take a back seat to more pressing concerns”
“Whatever biological imperatives this trip began with, they’re now taking a back seat to male pride”

There aren’t any idiomatic uses of put * the backseat in the COCA so I don’t think the idiom you thought you were trying to use is one that is in common use.

If you are speaking literally about placing objects or people into the back seat of a vehicle, you can use in, on, into, or onto, depending on the circumstance. You can only use into and onto for motions, while in and on can be used for either state or motion. Generally, you use in and into for people, and on and onto for objects, although sometimes people are on a backseat, though typically only if they are not sitting normally. Also, both back seat and backseat appear to be in equally common use. Examples from COCA:

In examples:

“He takes the sleeping child and places him in the backseat for the woman.”
“When riding in the backseat with two other people, you may sit on either side but never on the ‘hump.’”
“They drove him twenty miles to a hospital, my mother sitting in the back seat with the old man’s head in her lap.”

On examples:

“You can disassemble the tripod and place it on the backseat of your car.”
“Roy dropped a bag on the back seat.”
“If he got lucky and made it there with a lot of time to spare, his laptop was on the back seat, and he had a ton of paperwork to catch up on.”
“I took Alice along to sit on the back seat and steady those boxes all the way to Greenville”

Into examples:

“She climbed into the back seat and the car rolled forward and stopped again.”
“The woman picked up a woven basket and hopped into the back seat.”
“He envied the young officer who had pushed the guy into the back seat and slammed the door shut before reporting in on the radio”

Onto examples:

“She tucked the phone more securely into a pocket of her purse, unlocked the door, and slid the cat carrier onto the backseat.”
“Perhaps the detective imagined her thrown onto the back seat after being made unconscious in some manner.”
“Then Dad and I hurried out to load my Washington Stars onto the back seat of the Studebaker.”
“The young thin guy tossed his bag onto the back seat and slid right in after it.”
“Larry settled Alison onto the backseat, climbed in behind the steering wheel, and sat, staring through the windshield. He didn’t speak; Alison was deep in exhausted sleep.”

This is a question that well illustrates the inconsistency of analyses surrounding these whotsits.

From UsingEnglish.com

Phrasal Verb: Put on {Separable (optional [except with pronouns])}

Meaning: Start wearing

Example: I PUT my coat ON [You should put some clothes on.] [Put it on.]

(choosing the obviously corresponding usage from those given)

But there is no corresponding entry for 'rub on'.

I'd say that there is exactly the same degree of cohesiveness (unitariness) between rub and on in say 'Rub some lotion on' as there is between put and on in 'Put some clothes on'. (Some would class these particles as 'intransitive prepositions', perhaps to dodge the 'MWV or not?' issue.)

An argument that they should be considered as MWV (multi-word verb) + object constructions rather than simplex verb + PP constructions is the availability of:

Put on your coat. *Put on it.

Rub on the lotion. *Rub on it.

Sit on the chair. Sit on it.

Another is the availability of simplex equivalents: 'Don your coat' / 'Apply the lotion' / *'Asseat the chair'.

However, there are grey areas in the {MWV + DO} vs {V + PP} classification debate. This one is one of the trickier ones.

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'whotsits'? I know of at least 3 conflicting usages of 'phrasal verb'. The essential element is: 'Are rub on / take care of etc cohesive enough to be regarded as single lexemes?'