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I'm not sure which statement is more correct.

John has been with the team since 2010 and is currently a senior researcher

OR

John has been with the team since 2010 and currently is a senior researcher

(Hint: John has been with the team for sometime now and has been promoted few times having a much senior title now)

Word Choice

The first hurdle in answering your question is to decide whether you should use the word at all:

Caution: Use of currently, now, or presently is wordy when the verb it modifies is in the present tense. Not this: Sue is presently working at CBS. But this: Sue is working at CBS. — Martin Steinmann, Grammar Without Grief: The Ultimate A to Z for the Stylistically Clueless and the Grammatically Challenged, 1997.

The adverb currently is almost always unnecessary. It usually just restates information already conveyed through verb tenses and can be dropped with no loss of meaning. … But currently can be useful when contrasting current conditions with past or future conditions. — “How to use currently correctly,” Grammarist.com.

One writer won’t even make that concession:

Currently is my pet peeve. Yes, I realize that having a grammar pet peeve makes me a huge dork, and I’m okay with that. Currently is always redundant. You never need it.You’re not currently working for a law firm, you are working for a law firm. If you’re working there, it’s obviously currently.

The only time I can see this word and not groan is when the writer gives us information about the past, and then uses “currently” to transition to now. “I used to work for the mayor, but I’m currently working for the president.” Except you know what would sound better? Using “now” instead. “I used to work for the mayor, but now I work for the president.” So I guess there’s never a time when “currently” doesn’t make me groan. Because even when you need a transition word, “now” works better. — Alexis Grant, “Self-editing: 10 ways to tighten your copy,” 13 Dec. 2010.

Another writer, however, urges the use of currently instead of presently :

Usage panels, however, discourage the use of presently and urge the use of currently to describe something in the current moment. Ed Good, “Grammar Tips,” Grammar.com.

This, I suppose, is substituting something merely bad for something worse.

Frequency and Usage

If presently doesn’t enter the mix and, disregarding Ms. Grant et al., you decide currently is the word for you, then the question of word order arises.

A Google NGram suggests that English speakers overwhelmingly prefer is currently to currently is:

An NGrab only measures frequency: it cannot tell you whether the collocation is ungrammatical or non-standard. What the NGram does show, however, is that even though the word first appeared in the 1570s, frequnt use of currently is remarkably current. Note the steep incline beginning around 1965.

I suspect that this more frequent usage goes hand in hand with the resume, taken from the French résumé as a replacement for the more discursive application letter, and the emergence of the trade paperback in the 50s and 60s with its obligatory bio on the back cover which informs the reader that the author “currently resides in the Hamptons.” If one were to subtract all the resume-like hits in Google Books, the frequency would likely return to what it was in the 1930s.

ESL “Rules”

Teachers of English as a foreign language often substitute frequency for grammar, i.e., since most people say x, then x is the rule. This is both convenient and efficient, for trying to explain how and when native speakers don’t follow the usual pattern, particularly in more complex sentences, is as hopeless as trying to explain baseball or cricket.

For instance, a German site proclaims

If the verb is a form of »to be«, the adverb comes after the form of »to be«.

Aren’t those German quotes decorative? This rule is echoed by a site for Czech learners, while MyEnglishPages and CoLanguage, both international sites, restrict the rule to adverbs of frequency, as does this Polish site.

If your goal is to have students produce simple sentences such as the ones on the German site:

Mary is often at home.

Chris is sometimes nervous before tests.

in which there is no reason to deviate from meat-and-potatoes word order, then such a rule is appropriate. The problem, of course, is that it’s demonstrably false, even for adverbs of frequency:

The result is an 11‐hour day for Billy, one in which he seldom is at home during daylight hours. — “Gasoline Shortages Are Forcing Exurbanites to Readjust Their Life‐Style,” New York Times, 7 Feb. 1974.

The former elementary school teacher, who became a stay-at-home mom when her first child was born, seldom is at home, with all of the running-around she does with her kids. — Kellie B. Gormly, “High-mileage moms ,” TribLIVE (Pittsburgh Tribune), 24 Apr. 2007.

Graphical systems sometimes are used in this textbook to improve the learning process for students.— Michael Pidwirny, Part 1. Introduction to Physical Geography, 2018.

In the first two examples, the adverb precedes the verb because it is stressed: seldom doesn’t take its usual place because neither Billy nor the schoolteacher are able to spend much time at home, contrary to what one would expect. In the third example, the word order seems more like a stylistic choice. None, however, could be considered remotely ungrammatical.

In an elliptical construction, the adverb must come before the verb except in very informal speech:

There will be food in the shops, medical supplies in the hospitals, water in the taps and order on the streets (as much as there usually is). — David Bennun, “Brace yourself, Britain. Brexit is about to teach you what a crisis actually is,” The Guardian, 30 Nov. 2018.

And in indirect questions:

Bauer asked where it currently was and Fenske said it was in Cascade. — Minutes, Plan Commission Meeting, Town of Sheboygan, 7 Aug. 2018.

The usual word order can also be changed to eliminate ambiguity:

Her major research interest currently is changing family roles and consequences for the social structure. — Marvin Bernard Sussman, Betty E. Cogswell, eds., Cross-National Family Research, 1972, 223.

Using an adjective yields the rather ponderous her current major research interest. If the writer chooses not to begin the sentence with the adverb, which might give it too much weight, placing it after the verb could lead the reader to parse changing not as an adjective, but as a gerund, suggesting that she is the one doing the changing. Realizing that absurdity, the reader must reparse the sentence to figure out that family roles are changing instead.

Conclusion

So where does that leave you? With Alexis Grant, one might suggest a simple now. After all, why sound like every bio, book blurb, and resume on the planet? Either now or currently before the verb would stress the contrast to 2010, but no adverb can express what you’ve explained: that after a series of promotions he now occupies a much higher position. Adverbs can only modify the upward motion you want, not express it. In that case, a verb such as advanced (to the position of) or attained (the position of) might suit your purposes better.

I do not see any difference at all between the two sentences. they are both 'correct'. Neither is better than the other.

Only the first sentence is grammatically correct because there's a grammar rule to use adverbs after the verb 'to be'.

See the examples from Merriam-Webster's Dictionary:

currently 

: at the present time

// a product that is not currently available.

// Similar legislation is currently working successfully in 17 other states as well as in the nation's capital.

Similar examples from Cambridge Dictionary:

• The manufacturers are currently  testing  the new engine.




• The England cricket  team is currently on tour in Pakistan.

Some more examples from Oxford Living Dictionary:

‘The EC is currently attempting greater economic integration’

‘The land is currently used for turf cutting and for the commercial storage of caravans.’

‘He hopes to use the existing entrance to the land which is currently agricultural.’

‘The talented youngster is currently learning to play the keyboard and is a keen dancer.’

‘I am currently looking for an answer to ensure the staff and kids are looked after.’

‘The couple are currently on honeymoon in Cyprus and will be returning in a week's time.’

‘So far there are five injunctions in place but more are currently being processed.’

‘He said this could be achieved by allowing crews to do overtime which is currently banned.’

‘The biggest swarm of locusts in a decade is currently devouring crops in West Africa.’

‘She is currently working on her third novel that will be partly based in the North of England.’

‘The road to the Planks is currently shut, and people cannot walk or park their cars there.’

‘The service is currently aimed at men, but is set to expand into the ladies' market soon.’

‘Inflation is currently at its lowest in living memory and has been for years now.’

‘He is currently trying to design and build a vehicle to land on the surface of the planet Mercury.’

‘All of the books mentioned here are currently in print and available from good book shops.’

‘There is currently no suggestion that he was involved in illegal trading in drugs.

Cambridge Dictionary says that both statements are grammatically correct.

Examples:

He currently is directing TV sitcoms.

The Director is currently having talks in the US

Also, you can look at this link for many examples of the use of "currently is a".