Can a person be operating an aircraft in a "Careless" and "Reckless" manner at the same time? Or are these two different standards?
From a legal perspective, 'careless' and 'reckless' sound similar but are quite different.
Careless is the antonym of prudent and careful. There is usually no attempt to analyze the state of mind of the offender (other than in defense to reduce a more serious charge to the less serious careless.)
An example might be failing to lower the landing gear because the pilot forgot -- this would very likely fit the definition of careless. Careless is usually not a criminal violation in American legal frameworks (such as careless driving) but may be when combined with a statute violation.
Careless is also used as a description of the violation of the standard of care that is a reasonable expectation of care from those acting in a professional capacity or while subjecting others to danger or under a common carrier standard of care. The FAA use of careless is purposely ambiguous to allow any fitting definition to be used.
Reckless involves the wanton or willful disregard of other people's safety and/or property. It is usually a crime. Flying under a bridge or buzzing your buddies house would likely fit the definition of reckless. The burden of proof is on the prosecution to prove that state of mind of the defendant. HERE is a legal definition. HERE is the FAA's regulatory definition of what the FAA/ NTSB consider reckless.
Negligent would usually be considered between Careless and Reckless in severity and is usually the lowest level of criminal culpability. (Recall that FBI Director Comey's conclusion about Secretary Clinton's private email server rested on his definition of gross negligence vs extremely careless. The first is a crime with classified material; the second is not.)
The FAA likely uses Careless or Reckless in §91.13 to lower the bar that neither is tolerated and the difference between them will be dealt with in administration and penalties. It is a broad catch-all that includes negligent. If you go to the NTSB database of infractions, 91.13 is the most common. It is usually an add-on or derivative charge allowing for emergency revocation of a pilot's certificate. The applicable difference between the ALJ applying careless as opposed to reckless is the degree of penalty (license suspension vs license revocation) and, as you pointed out, whether the ARSP exemption to penalties apply.
Some examples of 91.13(a)claims; some of which parse through the difference of careless and reckless:

A pilot forgot to remove the nose-wheel chock of a propeller aircraft. He asked his wife to remove the chock from the aircraft with the engine running. She was fatally injured. CASE

Pilot wrongly declared a minimum fuel condition to ATC. CASE

'Buzzing' twice over a nude beach with a pilot on the beach as a witness. CASE

Misunderstood ATC clearance causes breach of separation. CASE

Helicopter pilot having sex with a passenger recorded on video. Martz case and ARTICLE


The Martz decision combined with the FAA definition of reckless can be derived into a standard of conduct that will likely ground you.
Under Martz, and previous cases, the standard of a finding careless or reckless is:

Regardless, as we have previously held, the Administrator need not
establish actual danger, a flight suddenly in peril, in order to
prevail in proving that a respondent has operated an aircraft in a
careless or reckless manner. In Administrator v. Lorenz, NTSB Order
No. EA-5205 at 2-3 (2006), we recognized that a showing of potential
endangerment is sufficient to prove a violation of § 91.13(a), and
cited several cases holding that proof of actual danger is unnecessary
for a § 91.13(a) charge.

And the relevant FAA standard is:

The inspector can infer a deliberate and willful disregard of the regulations or safety standards from the circumstances surrounding a violation. [omited text]
While there is no regulatory definition of the term, “reckless,” it
has been defined in cases decided by the NTSB. A reckless operation
results from the operation of an aircraft conducted with a deliberate
or willful disregard of the regulations or accepted standards of
safety so as to endanger the life or property of another either
potentially or actually. Accordingly, any such reckless behavior
violates § 91.13.

So potential of danger (not actual danger) that with a state of mind that can be inferred from the circumstances (not actual mens rea) is the applicable standard under § 91.13.
An excellent history of the development of 91.13 can be found HERE starting on p. 71 of the PDF.
Also keep in mind that the FAA is only dealing with administrative penalties; if you do something considered reckless there may be additional federal, local, or international criminal penalties as well as civil liability.
So to answer your question of Can a person be operating an aircraft in a “Careless” and “Reckless” manner at the same time? Or are these two different standards?
Since the spread between careless on the one hand and reckless on the other is large, one infraction is unlikely to be both simultaneously1. It is really more of an ambiguous spectrum of stupid and careless on the one extreme, to reckless and perhaps homicidal on the other extreme, with everything-we-don't-like in between. The difference between the two is largely the degree of the offense and the inferred state of mind of the offender.2

1 It very possible that an initial offense would be considered careless (such as forgetting to remove the nose chock) but the pilot's reaction is then judged to be reckless (sending his wife to remove the chock with a running engine resulting in her death.)
2Administrative legal procedures, such as FAA or NTSB enforcement actions, have significantly reduced burden of proof compared to criminal actions. See the FAA standard that allows intent to be inferred from the circumstances of the violation. See FSIM § 8900.1 CHG 422 14-3-5-1